1.787231650182965933013274890337008385337931402961810997784781470505557491750605686991077446567157288 Computed by Simon Plouffe with Maple using a dichotomic argument and fsolve in about 15 seconds. The source of the constant is the excellent essay of Steve Finch at http://pauillac.inria.fr/algo/bsolve/kom/kom.html " Jean-Paul Allouche and Michel Cosnard have recently proved, using a clever argument based on a classical theorem of Mahler, that q is transcendental."