Non-vanishing of the symmetric square <italic>L</italic>-function.

Abstract

We consider questions of non-vanishing of symmetric square <italic>L </italic>-functions lifted from Hecke cusp forms for the full modular group on the critical line in the weight aspect. We show that given any point on the critical line, for large enough even <italic>k</italic> there exists a Hecke cusp form <italic>f</italic> of weight <italic>k</italic> such that <italic> L</italic>(sym<super>2</super> <italic>f, s</italic>) is non-vanishing at that point. At the central point <italic>s</italic> = &frac12; we show that for a proportion of at least 1 - (1 + <italic>a</italic>)<super>-3 </super>, where 0 < <italic>a</italic> < &frac12;, of Hecke cusp forms <italic> f</italic> of weight less than <italic>K,</italic>, for large enough <italic> K,</italic> the value <italic>L</italic>(sym<super>2</super> <italic>f</italic>, &frac12;) &ne; 0. This same proportion has appeared in other works on <italic> L</italic>-functions belonging to the 'symplectic' family. The proportion 1 is conjectured.