BEHAVIOR OF AUTOMORPHIC L-FUNCTIONS AT THE POINTS s=1 AND s=1/2

摘要

Let S_k(N)~+ be the set of primitive cusp forms of even weight k for Γ_0(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~2 f), f ∈ S_2(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 forΣ_(f∈ S_k(1)~+) ω_fL(1/2, sym~2f) and Σ_(f∈S_k(1))ω_fL(1/2, f directX g) as k→∞are obtained.