ON THE DISTRIBUTION OF VALUES OF L(1, f)

摘要

Let S_k (N)~+ be the set of newforms of weight k for Γ_0 (N) , and let L(s, f) , f ∈ S_k (N)~+ , be the Hecke L -function of the form f. It is proved that for every integer m ≥ 1 , k = 2 , and N = p → ∞, Σ_(f ∈ S_2(N)~+) L~m(1, f) = 1/(12)B_mN + O(N~(1-α)), where B_m is a constant defined in the paper, and α = α (m) > 0 is a certain constant. This result implies the existence of the distribution function of the sequence {L(1,f), f ∈ S_2(N)~+}, N=p → ∞, and also yields an explicit expression for the corresponding characteristic function.