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

Let d be a fundamental discriminant, χd be the Dirichlet character associated to the quadratic field Q(d) and L(s,χd) be the Dirichlet L-function. In [Go], Goldfeld obtained an effective lower bound for L(1,χd) with uncalculated constants. For d<0, Oesterlé [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,χd) with d>0. Finally, as an application, we give an explicit lower bound for class numbers of certain real quadratic fields.