Behavior of Automorphic <Emphasis Type="Italic">L</Emphasis>-Functions at the Points s=1 and s=1/2

Let S_k(N)⁺ be the set of primitive cusp forms of even weight k for Γ₀(N) and let L(s, sym ²f) be the symmetric square L-function L(s, f) of a form f ∈ S_k(N)⁺. The moments of the variable L(1, sym ²f), f ∈ S₂(N)⁺, are computed for N = p, and the corresponding limiting distribution is determined in N-aspect. Let f ∈ S_k(1)⁺, g ∈ S_l(1)⁺, and ω_f = Γ(k - 1)/(4π)^k-1 〈f, f〉. Asymptotic formulas for and as k → ∞ are obtained. Bibliography: 17 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 302, 2003, pp. 149–167.