On the central value of symmetric square L-functions

Let S-k(N, chi) be the space of cusp forms of weight k, level N and character chi. For f epsilon S-k (N, chi) let L(s, sym(2) f) be the symmetric square L-function and L(s, f circle times f) be the Rankin-Selberg square attached to f. For fixed k >= 2, N prime, and real primitive chi, asymptotic formulas for the first and second moment of the central value of L(s, sym(2) f) and L(s, f circle times f) over a basis of S-k (N, chi) are given as N -> infinity. As an application it is shown that a positive proportion of the central values L(1/2, sym(2) f) does not vanish.