A few identities (III)

with Catalan

by

Simon Plouffe

Summer 1993

 

All equations are equal to 0 and where found using pari-Gp with an implementation of the LLL algorithm and an Interface with MapleV.

Identities : General case, gamma, Catalan, Zeta(3) and the Polygamma function

                                                       2
                       Psi(1, 5/6) + Psi(1, 1/6) - 4 Pi

                            _______________________

                            Li2(1/4) + Li2(1/3) - Li2(2/3)

                            _______________________

               8 Li2(1/6) + 8 Li2(5/6) + Psi(1, 5/6) - 5 Psi(1, 2/3)

                            _______________________

             - 32 G + 4 Psi(1, 1/4) - 3 Psi(1, 1/3) - 3 Psi(1, 2/3)

                            _______________________

                                                     2
                         3 Li2(1/4) + 3 Li2(3/4) - 1/2 Pi

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                    8 G + Psi(1, 3/4) - 6 Li2(1/6) - 6 Li2(5/6)

                            _______________________
                                                       2
                       Psi(1, 1/4) + Psi(1, 3/4) - 2 Pi

                            _______________________

                            Li2(1/4) + Li2(1/3) - Li2(2/3)

                            _______________________

                                    2
          - 8 G + Psi(1, 1/4) + 3 Pi  - 3 Psi(1, 1/3) - 3 Psi(1, 2/3)

                            _______________________

             - 32 Li2(1/4) - 32 Li2(3/4) + Psi(1, 1/6) + 5 Psi(1, 2/3)

                            _______________________

                                           2
                        - 3 Li2(1/4) - 1/2 Pi  + 6 Li2(2/3)

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                  - Psi(1, 5/6) - Psi(1, 1/3) + 4 Psi(1, 2/3)

                            _______________________

                                         2
                     3 Psi(1, 1/6) + 4 Pi  - 15 Psi(1, 1/3)

                            _______________________

          - 2 Psi(1, 1/4) - 2 Psi(1, 3/4) + Psi(1, 5/6) + Psi(1, 1/6)

                            _______________________

                                                   2
                           - 8 G + Psi(1, 1/4) - Pi

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                8 Li2(1/6) + 8 Li2(5/6) - Psi(1, 1/3) - Psi(1, 2/3)

                            _______________________

                   Psi(1, 5/6) + Psi(1, 1/3) - 4 Psi(1, 2/3)

                            _______________________

                                                       2
                       Psi(1, 5/6) + Psi(1, 1/6) - 4 Pi

                            _______________________

                            Li2(1/4) + Li2(1/3) - Li2(2/3)

                            _______________________

      16 G + Psi(1, 1/4) + 3 Psi(1, 3/4) + 3 Psi(1, 1/6) - 15 Psi(1, 1/3)

                            _______________________

                                                       2
                       Psi(1, 5/6) + Psi(1, 1/6) - 4 Pi

                            _______________________

                                2
                        - 1/2 Pi  + 3 Li2(1/3) + 3 Li2(2/3)

                            _______________________

                                                       2
                       Psi(1, 5/6) + Psi(1, 1/6) - 4 Pi

                            _______________________

               Psi(1, 1/4) + Psi(1, 3/4) + 12 Li2(1/4) - 24 Li2(2/3)

                            _______________________

                                          2
                         3 Li2(1/4) - 1/2 Pi  + 6 Li2(1/3)

                            _______________________

               24 Li2(1/4) + Psi(1, 5/6) + Psi(1, 1/6) - 48 Li2(2/3)

                            _______________________

                                         2
          - Li2(1/4) + Psi(1, 5/6) + 3/2 Pi  - 5 Psi(1, 2/3) - 2 Li2(1/3)

                            _______________________

                    8 G - Psi(1, 1/4) + 6 Li2(1/6) + 6 Li2(5/6)

                            _______________________

              - 8 Li2(1/6) - 8 Li2(5/6) - Psi(1, 5/6) + 5 Psi(1, 2/3)

                            _______________________

                                           2
                        - 3 Li2(1/4) - 1/2 Pi  + 6 Li2(2/3)

                            _______________________

                       Li2(1/6) + Li2(5/6) - Li2(1/3) - Li2(2/3)

                            _______________________

                                                       2
                       Psi(1, 1/4) + Psi(1, 3/4) - 2 Pi

                            _______________________

                                                       2
                       Psi(1, 1/4) + Psi(1, 3/4) - 2 Pi

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                                          2
                         3 Li2(1/4) - 1/2 Pi  + 6 Li2(1/3)

                            _______________________

                               2
                         1/2 Pi  - 3 Li2(1/3) - 3 Li2(2/3)

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

               24 Li2(1/4) + Psi(1, 5/6) + Psi(1, 1/6) - 48 Li2(2/3)

                            _______________________

              32 Li2(1/6) + 32 Li2(5/6) - Psi(1, 1/6) - 5 Psi(1, 2/3)

                            _______________________

                   Psi(1, 5/6) + Psi(1, 1/3) - 4 Psi(1, 2/3)

                            _______________________

                       Li2(1/6) + Li2(5/6) - Li2(1/3) - Li2(2/3)

                            _______________________

                                          2
                   - 3 Psi(1, 1/6) + 16 Pi  - 15 Psi(1, 2/3)

                            _______________________

                                                      2
                        - 3 Li2(1/6) - 3 Li2(5/6) + 1/2 Pi

                            _______________________

                                                      2
                        - 3 Li2(1/4) - 3 Li2(3/4) + 1/2 Pi

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                   Psi(1, 1/6) - 4 Psi(1, 1/3) + Psi(1, 2/3)

                            _______________________

                                                       2
                       Psi(1, 5/6) + Psi(1, 1/6) - 4 Pi

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                         - 2 Li2(1/4) - Li2(3/4) + 2 Li2(2/3)

                            _______________________

                       Li2(1/6) + Li2(5/6) - Li2(1/3) - Li2(2/3)

                            _______________________

                         2
                     4 Pi  - 3 Psi(1, 1/3) - 3 Psi(1, 2/3)

                            _______________________

                   Psi(1, 5/6) + Psi(1, 1/3) - 4 Psi(1, 2/3)

                            _______________________

                                         2
                          - 3 Li2(3/4) + Pi  - 6 Li2(2/3)

                            _______________________

                                          2
                         3 Li2(1/4) + 1/2 Pi  - 6 Li2(2/3)

                            _______________________

                   Psi(1, 5/6) + Psi(1, 1/3) - 4 Psi(1, 2/3)

                            _______________________

                                                       2
                       Psi(1, 5/6) + Psi(1, 1/6) - 4 Pi

                            _______________________

                            Li2(1/4) + Li2(1/3) - Li2(2/3)

                            _______________________

                    8 G + Psi(1, 3/4) + 6 Li2(1/4) - 12 Li2(2/3)

                            _______________________

                                          2
                         3 Li2(1/4) - 1/2 Pi  + 6 Li2(1/3)

                            _______________________

                        16 G - Psi(1, 1/4) + Psi(1, 3/4)

                            _______________________

                                                   2
                           - 8 G + Psi(1, 1/4) - Pi

                            _______________________

          - 2 Psi(1, 1/4) - 2 Psi(1, 3/4) + Psi(1, 5/6) + Psi(1, 1/6)

                            _______________________

               Psi(1, 1/4) + Psi(1, 3/4) - 12 Li2(1/6) - 12 Li2(5/6)

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                                              2
                      Li2(1/4) - Li2(3/4) + 1/2 Pi  - 4 Li2(2/3)

                            _______________________

                       - 16 G + Psi(1, 1/4) - Psi(1, 3/4)

                            _______________________

               8 Li2(1/6) + 8 Li2(5/6) + Psi(1, 5/6) - 5 Psi(1, 2/3)

                            _______________________

                    8 G - Psi(1, 1/4) - 6 Li2(1/4) + 12 Li2(2/3)

                            _______________________

                   Psi(1, 1/6) - 4 Psi(1, 1/3) + Psi(1, 2/3)

                            _______________________

                                           2
                        - 3 Li2(1/4) - 1/2 Pi  + 6 Li2(2/3)

                            _______________________

                    Li2(3/4) - 2 Li2(1/6) - 2 Li2(5/6) + 2 Li2(2/3)

                            _______________________

                            Li2(1/4) + Li2(1/3) - Li2(2/3)

                            _______________________

                            Li2(1/4) + Li2(1/3) - Li2(2/3)

                            _______________________

                   Psi(1, 5/6) + Psi(1, 1/3) - 4 Psi(1, 2/3)

                            _______________________

                    8 G + Psi(1, 3/4) - 6 Li2(1/6) - 6 Li2(5/6)

                            _______________________

                                                      2
                        - 3 Li2(1/4) - 3 Li2(3/4) + 1/2 Pi

                            _______________________

                                                      2
                        - 3 Li2(1/6) - 3 Li2(5/6) + 1/2 Pi

                            _______________________

              - 8 Li2(1/4) - Psi(1, 5/6) + 5 Psi(1, 2/3) - 16 Li2(1/3)

                            _______________________

                                                     2
                         3 Li2(1/6) + 3 Li2(5/6) - 1/2 Pi

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                   - 8 G + Psi(1, 1/4) - 6 Li2(1/4) - 12 Li2(1/3)

                            _______________________

            128 G - 16 Psi(1, 1/4) + 3 Psi(1, 1/6) + 15 Psi(1, 2/3)

                            _______________________

             - 32 Li2(1/4) - 32 Li2(3/4) + Psi(1, 5/6) + 5 Psi(1, 1/3)

                            _______________________

                                                       2
                       Psi(1, 5/6) + Psi(1, 1/6) - 4 Pi

                            _______________________

                                          2
                         3 Li2(1/4) + 1/2 Pi  - 6 Li2(2/3)

                            _______________________

                   Psi(1, 5/6) + Psi(1, 1/3) - 4 Psi(1, 2/3)

                            _______________________

                         2
                     4 Pi  - 3 Psi(1, 1/3) - 3 Psi(1, 2/3)

                            _______________________

                                                       2
                       Psi(1, 5/6) + Psi(1, 1/6) - 4 Pi

                            _______________________

                                          2
                         3 Li2(1/4) + 1/2 Pi  - 6 Li2(2/3)

                            _______________________

                         - 2 Li2(1/4) - Li2(3/4) + 2 Li2(2/3)

                            _______________________

                                                      2
                        - 3 Li2(1/4) - 3 Li2(3/4) + 1/2 Pi

                            _______________________

                                   2
           Li2(1/6) + Li2(5/6) + 5/2 Pi  - 2 Psi(1, 1/3) - 2 Psi(1, 2/3)

                            _______________________

               Psi(1, 1/4) + Psi(1, 3/4) - 12 Li2(1/4) - 24 Li2(1/3)

                            _______________________

                                                       2
                       Psi(1, 1/4) + Psi(1, 3/4) - 2 Pi

                            _______________________

                            Li2(1/4) + Li2(1/3) - Li2(2/3)

                            _______________________

                            Li2(1/4) + Li2(1/3) - Li2(2/3)

                            _______________________

                   - 8 G + Psi(1, 1/4) - 6 Li2(1/4) - 6 Li2(3/4)

                            _______________________

                                                  2
                            8 G + Psi(1, 3/4) - Pi

                            _______________________

      16 Li2(1/6) + 16 Li2(5/6) - Psi(1, 5/6) - 2 Psi(1, 1/6) + 5 Psi(1, 1/3)

                            _______________________

               8 Li2(1/6) + 8 Li2(5/6) + Psi(1, 1/6) - 5 Psi(1, 1/3)

                            _______________________

                   Psi(1, 5/6) + Psi(1, 1/3) - 4 Psi(1, 2/3)

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                                          2
                   - 3 Psi(1, 5/6) + 16 Pi  - 15 Psi(1, 1/3)

                            _______________________

                   Psi(1, 5/6) + Psi(1, 1/3) - 4 Psi(1, 2/3)

                            _______________________

                         - 2 Li2(1/4) - Li2(3/4) + 2 Li2(2/3)

                            _______________________

              - 8 Li2(1/6) - 8 Li2(5/6) - Psi(1, 5/6) + 5 Psi(1, 2/3)

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

              16 Li2(3/4) - Psi(1, 1/6) - 5 Psi(1, 2/3) + 32 Li2(2/3)

                            _______________________

                  Psi(1, 5/6) + 4 Psi(1, 1/6) - 15 Psi(1, 1/3)

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                   Psi(1, 5/6) + Psi(1, 1/3) - 4 Psi(1, 2/3)

                            _______________________

                      - Li2(1/6) - Li2(5/6) + Li2(1/3) + Li2(2/3)

                            _______________________

               - 8 Li2(1/6) - 8 Li2(5/6) + Psi(1, 1/3) + Psi(1, 2/3)

                            _______________________

             32 G + 4 Psi(1, 3/4) + 3 Psi(1, 1/6) - 15 Psi(1, 1/3)

                            _______________________

    - Psi(1, 1/4) - Psi(1, 3/4) - 4 Li2(1/4) - 2 Psi(1, 5/6) + 10 Psi(1, 2/3)

         - 8 Li2(1/3)

                            _______________________

              - 8 Li2(1/4) + Psi(1, 5/6) - 5 Psi(1, 2/3) + 16 Li2(2/3)

                            _______________________

                               2
                         1/2 Pi  - 3 Li2(1/3) - 3 Li2(2/3)

                            _______________________

                       Li2(1/4) + Li2(3/4) - Li2(1/6) - Li2(5/6)

                            _______________________

                                         2
                          - 3 Li2(3/4) + Pi  - 6 Li2(2/3)

                            _______________________

                                                      2
                        - 3 Li2(1/4) - 3 Li2(3/4) + 1/2 Pi

                            _______________________

                         - 2 Li2(1/4) - Li2(3/4) + 2 Li2(2/3)

                            _______________________

                    8 G + Psi(1, 3/4) - 6 Li2(1/4) - 12 Li2(1/3)

                            _______________________

             - 32 G + 4 Psi(1, 1/4) - 3 Psi(1, 1/3) - 3 Psi(1, 2/3)

                            _______________________

                            Li2(1/4) + Li2(1/3) - Li2(2/3)

                            _______________________

        - 2 Psi(1, 1/4) - 2 Psi(1, 3/4) + 3 Psi(1, 1/3) + 3 Psi(1, 2/3)

                            _______________________

              - 8 Li2(1/4) + Psi(1, 1/6) - 5 Psi(1, 1/3) + 16 Li2(2/3)

                            _______________________

                               2
                         1/2 Pi  - 3 Li2(1/3) - 3 Li2(2/3)

                            _______________________

                                          2
                    - 3 Psi(1, 1/6) - 4 Pi  + 15 Psi(1, 1/3)

                            _______________________

                      Li2(1/4) - Li2(1/6) - Li2(5/6) + 2 Li2(1/3)

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                   Psi(1, 5/6) + Psi(1, 1/3) - 4 Psi(1, 2/3)

                            _______________________

                         2
                     4 Pi  - 3 Psi(1, 1/3) - 3 Psi(1, 2/3)

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

              - 24 Li2(1/4) - 24 Li2(3/4) + Psi(1, 5/6) + Psi(1, 1/6)

                            _______________________

                                         2
                    3 Psi(1, 5/6) - 16 Pi  + 15 Psi(1, 1/3)

                            _______________________

                   - 8 G - Psi(1, 3/4) + 6 Li2(1/6) + 6 Li2(5/6)

                            _______________________

              - 12 Li2(3/4) + Psi(1, 5/6) + Psi(1, 1/6) - 24 Li2(2/3)

                            _______________________

                       Li2(1/6) + Li2(5/6) - Li2(1/3) - Li2(2/3)

                            _______________________

                    8 G - Psi(1, 1/4) + 3 Li2(3/4) + 6 Li2(2/3)

                            _______________________

                         2
                     4 Pi  - 3 Psi(1, 1/3) - 3 Psi(1, 2/3)

                            _______________________

               - Psi(1, 1/4) - Psi(1, 3/4) + 6 Li2(3/4) + 12 Li2(2/3)

                            _______________________

                 - 4 Psi(1, 5/6) - Psi(1, 1/6) + 15 Psi(1, 2/3)

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                                                      2
                        - 3 Li2(1/6) - 3 Li2(5/6) + 1/2 Pi

                            _______________________

                                                       2
                       Psi(1, 1/4) + Psi(1, 3/4) - 2 Pi

                            _______________________

                                     2
             Li2(1/4) + Li2(3/4) - 3/2 Pi  + Psi(1, 1/3) + Psi(1, 2/3)

                            _______________________

                    8 G - Psi(1, 1/4) + 6 Li2(1/3) + 6 Li2(2/3)

                            _______________________

                                                   2
                           - 8 G + Psi(1, 1/4) - Pi

                            _______________________

                   Psi(1, 5/6) + Psi(1, 1/3) - 4 Psi(1, 2/3)

                            _______________________

                                                    2
               8 G + Psi(1, 3/4) + 3 Li2(1/4) - 3/2 Pi  + 6 Li2(1/3)

                            _______________________

                             1/2
                         Pi 3    + Psi(1/6) - Psi(5/6)

                            _______________________

                   Psi(1, 5/6) + Psi(1, 1/3) - 4 Psi(1, 2/3)

                            _______________________

                                                      2
                        - 3 Li2(1/4) - 3 Li2(3/4) + 1/2 Pi

                            _______________________

                   Psi(1, 1/6) - 4 Psi(1, 1/3) + Psi(1, 2/3)

                            _______________________

               - Psi(1, 1/3) - Psi(1, 2/3) + 8 Li2(1/3) + 8 Li2(2/3)

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                         - 2 Li2(1/4) - Li2(3/4) + 2 Li2(2/3)

                            _______________________

                        16 G - Psi(1, 1/4) + Psi(1, 3/4)

                            _______________________

                                2
                        - 1/2 Pi  + 3 Li2(1/3) + 3 Li2(2/3)

                            _______________________

                        16 G - Psi(1, 1/4) + Psi(1, 3/4)

                            _______________________

                  - Psi(1, 5/6) - Psi(1, 1/3) + 4 Psi(1, 2/3)

                            _______________________

                                           2
                        - 3 Li2(1/4) + 1/2 Pi  - 6 Li2(1/3)

                            _______________________

                   - 8 G - Psi(1, 3/4) + 6 Li2(1/3) + 6 Li2(2/3)

                            _______________________

                         - 2 Li2(1/4) - Li2(3/4) + 2 Li2(2/3)

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                                           2
                        - 3 Li2(1/4) + 1/2 Pi  - 6 Li2(1/3)

                            _______________________

                                                     2
                         3 Li2(1/4) + 3 Li2(3/4) - 1/2 Pi

                            _______________________

                                                      2
                        - 3 Li2(1/6) - 3 Li2(5/6) + 1/2 Pi

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

               - 32 G + 4 Psi(1, 1/4) - Psi(1, 5/6) - Psi(1, 1/6)

                            _______________________

                                                   2
                           - 8 G + Psi(1, 1/4) - Pi

                            _______________________

                            Li2(1/4) + Li2(1/3) - Li2(2/3)

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                                   2
           8 G + Psi(1, 3/4) - 5 Pi  + 3 Psi(1, 1/3) + 3 Psi(1, 2/3)

                            _______________________

              - Psi(1, 1/4) - Psi(1, 3/4) + 12 Li2(1/6) + 12 Li2(5/6)

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                         2
                     4 Pi  - 3 Psi(1, 1/3) - 3 Psi(1, 2/3)

                            _______________________

                              - Li2(3/4) + 2 Li2(1/3)

                            _______________________

                          2
                    - 4 Pi  + 3 Psi(1, 1/3) + 3 Psi(1, 2/3)

                            _______________________

              - 8 Li2(1/6) - 8 Li2(5/6) - Psi(1, 5/6) + 5 Psi(1, 2/3)

                            _______________________

                                       2
       Psi(1, 1/4) + Psi(1, 3/4) + 2 Pi  - 3 Psi(1, 1/3) - 3 Psi(1, 2/3)

                            _______________________

                                           2
                        - 3 Li2(1/4) + 1/2 Pi  - 6 Li2(1/3)

                            _______________________

                                                   2
                           - 8 G + Psi(1, 1/4) - Pi

                            _______________________

               Psi(1, 1/4) + Psi(1, 3/4) - 12 Li2(1/4) - 12 Li2(3/4)

                            _______________________

             - 16 Li2(3/4) + Psi(1, 5/6) + 5 Psi(1, 1/3) - 32 Li2(2/3)

                            _______________________

               4 Li2(3/4) + Psi(1, 1/6) - 5 Psi(1, 1/3) + 8 Li2(2/3)

                            _______________________

                              1/2
                        - Pi 3    - Psi(1/6) + Psi(5/6)

                            _______________________

                                                       2
                       Psi(1, 5/6) + Psi(1, 1/6) - 4 Pi

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

               8 Li2(1/6) + 8 Li2(5/6) + Psi(1, 1/6) - 5 Psi(1, 1/3)

                            _______________________

                   - 8 G + Psi(1, 1/4) - 6 Li2(1/4) - 6 Li2(3/4)

                            _______________________

                   Psi(1, 5/6) + Psi(1, 1/3) - 4 Psi(1, 2/3)

                            _______________________

                                           2
                        - 3 Li2(1/4) + 1/2 Pi  - 6 Li2(1/3)

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                                                     2
                         3 Li2(1/6) + 3 Li2(5/6) - 1/2 Pi

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

     - 16 Li2(1/4) - 16 Li2(3/4) + Psi(1, 5/6) + 2 Psi(1, 1/6) - 5 Psi(1, 1/3)

                            _______________________

                                         2
                     3 Psi(1, 5/6) + 4 Pi  - 15 Psi(1, 2/3)

                            _______________________

                       - 16 G + Psi(1, 1/4) - Psi(1, 3/4)

                            _______________________

                                                  2
                            8 G + Psi(1, 3/4) - Pi

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                                                  2
                            8 G + Psi(1, 3/4) - Pi

                            _______________________

                             1/2
                         Pi 3    + Psi(1/6) - Psi(5/6)

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                                         2
                     3 Psi(1, 5/6) + 4 Pi  - 15 Psi(1, 2/3)

                            _______________________

                                                       2
                       Psi(1, 1/4) + Psi(1, 3/4) - 2 Pi

                            _______________________

                                                        2
                      - Psi(1, 1/4) - Psi(1, 3/4) + 2 Pi

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                   Psi(1, 5/6) + Psi(1, 1/3) - 4 Psi(1, 2/3)

                            _______________________

               - 32 G - 4 Psi(1, 3/4) + Psi(1, 5/6) + Psi(1, 1/6)

                            _______________________

                                                      2
                        - 3 Li2(1/6) - 3 Li2(5/6) + 1/2 Pi

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                  - Psi(1, 5/6) - Psi(1, 1/3) + 4 Psi(1, 2/3)

                            _______________________

             - 32 Li2(1/6) - 32 Li2(5/6) + Psi(1, 1/6) + 5 Psi(1, 2/3)

                            _______________________

                                           2
                        - 3 Li2(1/4) - 1/2 Pi  + 6 Li2(2/3)

                            _______________________

               8 Li2(1/6) + 8 Li2(5/6) + Psi(1, 5/6) - 5 Psi(1, 2/3)

                            _______________________

                                  1/2
     3 gamma + 12 ln(2) - 1/2 Pi 3    - 9/2 ln(3) + 3 Psi(1/4) + 3 Psi(3/4)

          - 2 Psi(1/6) - Psi(5/6)

                            _______________________

                         2
                     4 Pi  - 3 Psi(1, 1/3) - 3 Psi(1, 2/3)

                            _______________________

                   Psi(1, 5/6) + Psi(1, 1/3) - 4 Psi(1, 2/3)

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                                         2
                          - 3 Li2(3/4) + Pi  - 6 Li2(2/3)

                            _______________________

                            Li2(1/4) + Li2(1/3) - Li2(2/3)

                            _______________________

                   Psi(1, 5/6) + Psi(1, 1/3) - 4 Psi(1, 2/3)

                            _______________________

                    8 G - Psi(1, 1/4) + 6 Li2(1/4) + 12 Li2(1/3)

                            _______________________

                                        2
                           3 Li2(3/4) - Pi  + 6 Li2(2/3)

                            _______________________

                Psi(1, 1/4) + Psi(1, 3/4) - 6 Li2(3/4) - 12 Li2(2/3)

                            _______________________

                8 Li2(1/4) + Psi(1, 1/3) + Psi(1, 2/3) - 16 Li2(2/3)

                            _______________________

                                                   2
                           - 8 G - Psi(1, 3/4) + Pi

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

              - 24 Li2(1/4) - Psi(1, 5/6) - Psi(1, 1/6) + 48 Li2(2/3)

                            _______________________

                            Li2(1/4) + Li2(1/3) - Li2(2/3)

                            _______________________

                                          2
                    - 3 Psi(1, 1/6) - 4 Pi  + 15 Psi(1, 1/3)

                            _______________________

                       Li2(1/4) + Li2(3/4) - Li2(1/6) - Li2(5/6)

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                    8 G + Psi(1, 3/4) - 6 Li2(1/6) - 6 Li2(5/6)

                            _______________________

                                                      2
                        - 3 Li2(1/6) - 3 Li2(5/6) + 1/2 Pi

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                                2
                        - 1/2 Pi  + 3 Li2(1/3) + 3 Li2(2/3)

                            _______________________

         2 Psi(1, 1/4) + 2 Psi(1, 3/4) - 3 Psi(1, 1/3) - 3 Psi(1, 2/3)

                            _______________________

                  - Psi(1, 5/6) - Psi(1, 1/3) + 4 Psi(1, 2/3)

                            _______________________

                   Psi(1, 5/6) + Psi(1, 1/3) - 4 Psi(1, 2/3)

                            _______________________

                    8 G + Psi(1, 3/4) - 6 Li2(1/6) - 6 Li2(5/6)

                            _______________________

                                                   2
                           - 8 G - Psi(1, 3/4) + Pi

                            _______________________

                                                   2
                           - 8 G + Psi(1, 1/4) - Pi

                            _______________________

                 - Psi(1, 5/6) - 4 Psi(1, 1/6) + 15 Psi(1, 1/3)

                            _______________________

               8 Li2(1/4) + Psi(1, 5/6) - 5 Psi(1, 2/3) + 16 Li2(1/3)

                            _______________________

              - 24 Li2(1/4) - 24 Li2(3/4) + Psi(1, 5/6) + Psi(1, 1/6)

                            _______________________

                         - 2 Li2(1/4) - Li2(3/4) + 2 Li2(2/3)

                            _______________________

               Psi(1, 1/4) + Psi(1, 3/4) - 12 Li2(1/6) - 12 Li2(5/6)

                            _______________________

                                                   2
                           - 8 G - Psi(1, 3/4) + Pi

                            _______________________

                     - Li2(1/4) - Li2(1/6) - Li2(5/6) + 2 Li2(2/3)

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

               Psi(1, 1/4) + Psi(1, 3/4) - 12 Li2(1/4) - 12 Li2(3/4)

                            _______________________

                                                  2
                            8 G + Psi(1, 3/4) - Pi

                            _______________________

                         - 2 Li2(1/4) - Li2(3/4) + 2 Li2(2/3)
 
                            _______________________

                   Psi(1, 5/6) + Psi(1, 1/3) - 4 Psi(1, 2/3)

                            _______________________

                            Li2(1/4) + Li2(1/3) - Li2(2/3)

                            _______________________

                         - 2 Li2(1/4) - Li2(3/4) + 2 Li2(2/3)

                            _______________________

              32 Li2(1/6) + 32 Li2(5/6) - Psi(1, 5/6) - 5 Psi(1, 1/3)

                            _______________________

                         - 2 Li2(1/4) - Li2(3/4) + 2 Li2(2/3)

                            _______________________

                    8 G + Psi(1, 3/4) - 3 Li2(3/4) - 6 Li2(2/3)

                            _______________________

                    8 G + Psi(1, 3/4) - 6 Li2(1/4) - 6 Li2(3/4)

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                            Li2(1/4) + Li2(1/3) - Li2(2/3)

                            _______________________

                                          2
                         3 Li2(1/4) + 1/2 Pi  - 6 Li2(2/3)

                            _______________________

                            Li2(1/4) + Li2(1/3) - Li2(2/3)

                            _______________________

     8 G - Psi(1, 1/4) - 2 Li2(1/4) - Psi(1, 5/6) + 5 Psi(1, 2/3) - 4 Li2(1/3)

                            _______________________

                            Li2(1/4) + Li2(1/3) - Li2(2/3)

                            _______________________

                                                       2
                       Psi(1, 5/6) + Psi(1, 1/6) - 4 Pi

                            _______________________

                                                  2
                            8 G + Psi(1, 3/4) - Pi

                            _______________________

                          2
             - Li2(3/4) - Pi  + Psi(1, 1/3) + Psi(1, 2/3) - 2 Li2(2/3)

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                   Psi(1, 5/6) + Psi(1, 1/3) - 4 Psi(1, 2/3)

                            _______________________

                  Psi(1, 5/6) + 4 Psi(1, 1/6) - 15 Psi(1, 1/3)

                            _______________________

     - 8 Li2(1/4) + Psi(1, 5/6) + 5 Psi(1, 1/6) - 20 Psi(1, 1/3) + 16 Li2(2/3)

                            _______________________

                                                        2
                      - Psi(1, 1/4) - Psi(1, 3/4) + 2 Pi

                            _______________________

             32 G - 4 Psi(1, 1/4) - 3 Psi(1, 5/6) + 15 Psi(1, 2/3)

                            _______________________

                       Li2(1/4) + Li2(3/4) - Li2(1/6) - Li2(5/6)

                            _______________________

                                                       2
                       Psi(1, 5/6) + Psi(1, 1/6) - 4 Pi

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

              - 12 Li2(3/4) + Psi(1, 5/6) + Psi(1, 1/6) - 24 Li2(2/3)

                            _______________________

                    Li2(3/4) - 2 Li2(1/6) - 2 Li2(5/6) + 2 Li2(2/3)

                            _______________________

                         - 2 Li2(1/4) - Li2(3/4) + 2 Li2(2/3)

                            _______________________

              - 12 Li2(3/4) + Psi(1, 5/6) + Psi(1, 1/6) - 24 Li2(2/3)

                            _______________________

                         - 2 Li2(1/4) - Li2(3/4) + 2 Li2(2/3)

                            _______________________

                               Li2(3/4) - 2 Li2(1/3)

                            _______________________

                                                   2
                           - 8 G - Psi(1, 3/4) + Pi

                            _______________________

                                                       2
                       Psi(1, 1/4) + Psi(1, 3/4) - 2 Pi

                            _______________________

                                                       2
                       Psi(1, 1/4) + Psi(1, 3/4) - 2 Pi

                            _______________________

                         2
                     4 Pi  - 3 Psi(1, 1/3) - 3 Psi(1, 2/3)

                            _______________________

        - 8 Psi(1, 1/4) - 8 Psi(1, 3/4) + 3 Psi(1, 5/6) + 15 Psi(1, 1/3)

                            _______________________

                            Li2(1/4) + Li2(1/3) - Li2(2/3)

                            _______________________

                                                      2
                        - 3 Li2(1/4) - 3 Li2(3/4) + 1/2 Pi

                            _______________________

              - 8 Li2(1/4) + Psi(1, 5/6) - 5 Psi(1, 2/3) + 16 Li2(2/3)

                            _______________________

                                                       2
                       Psi(1, 1/4) + Psi(1, 3/4) - 2 Pi

                            _______________________

                   Psi(1, 5/6) + Psi(1, 1/3) - 4 Psi(1, 2/3)

                            _______________________

                                                          2 
                        - 3 Li2(1/4) - 3 Li2(3/4) + 1/2 Pi

                            _______________________

                                       2
       Psi(1, 1/4) + Psi(1, 3/4) + 2 Pi  - 3 Psi(1, 1/3) - 3 Psi(1, 2/3)

                            _______________________