Quantitative non-vanishing of central values of certain L-functions on GL(2) x GL(3)

Let phi be an even Hecke-Maass cusp form on SL2(Z) whose L-function does not vanish at the center of the functional equation. In this article, we obtain an exact formula of the average of triple products of phi, f and (f) over bar, where f runs over an orthonornial basis H-k of Hecke eigen elliptic cusp forms on SL2(Z) of a fixed weight k >= 4. As an application, we prove a quantitative non-vanishing results on the central values for the family of degree 6 L-functions L(s, phi x Ad f) with f in the union of H-k (K <= k < 2K) as K -> infinity.