An explicit lower bound for special values of Dirichlet L-functions

Let d be a fundamental discriminant, chi(d) be the Dirichlet character associated to the quadratic field Q(root d) and L(s, chi(d)) be the Dirichlet L-function. In [Go], Goldfeld obtained an effective lower bound for L(1, chi d) with uncalculated constants. For d < 0, Oesterle [Oe] computed the constants. However, for d > 0, the constants are not computed yet. In this paper, we compute the constants for d > 0 and give an explicit lower bound for L(1, chi(d)) with d > 0. Finally, as an application, we give an explicit lower bound for class numbers of certain real quadratic fields. (C) 2018 Elsevier Inc. All rights reserved.