Behavior of the L-functions of cusp forms ats=1

Let f(z) be a Hecke eigenform in the space S_2k(Γ) of holomorphic Γ-cusp forms of even weight 2k, Γ=SL(2,ℤ); let L_f(s) be the L-function of f(z). The goal of this paper is to obtain some results on L_f(1) as k increases. In particular, we prove an analogue of the classical Landau theorem in the theory of Dirichlet L-functions and (under a very plausible hypothesis) an analogue of the famous Siegel theorem. Bibliography: 15 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 204, 1993, pp. 37–54. Translated by E. P. Golubeva.