Explicit upper bounds for values at s=1 of Dirichlet L-series associated with primitive even characters

Let S be a given finite set of pairwise, distinct rational primes. We give an explicit constant kappa(S) such that for any even primitive Dirichlet character chi of conductor q(Z) > 1 we have \{(Π) under barpepsilonS (1 - (χ(p) under barp)}L(1,chi)\less than or equal to(1) under bar2{(Π) under barpepsilonS(1 - (1) under barp)} (log q(chi) + kappa(S)) + o(1), where o(1) is an explicit error term which tends rapidly to zero when q, goes to infinity. (C) 2003 Elsevier Inc. All rights reserved.