Abstract
A generalized Riemann hypothesis states that all zeros of the completed Hecke L-function L⁎(f,s) of a normalized Hecke eigenform f on the full modular group should lie on the vertical line Re(s)=k2. It was shown in [6] that there exists a Hecke eigenform f of weight k such that L⁎(f,s)≠0 for sufficiently large k and any point on the line segments Im(s)=t0,k−12<Re(s)<k2−ϵ,k2+ϵ<Re(s)<k+12, for any given real number t0 and a positive real number ϵ. This paper concerns the non-vanishing of the product L⁎(f,s)L⁎(f,w)(s,w∈C) on average.
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