In 1976, Zagier established a functional equation for the generalized Dirichlet L-functions that are part of the Fourier-Whittaker expansion of halfintegral weight Eisenstein series. The special values of these L-functions at 1/2 and at 1 are of particular interest because of the connection with the Selberg trace formula, with moments of symmetric square L-functions and with the prime geodesic theorem. In this talk, we describe various properties of Zagier L-functions and consider several problems related to the asymptotic evaluation of averages of special L-values.