The Central value of the Rankin–Selberg L-Functions

Abstract.

Let f be a Maass form for SL $(3, {\mathbb{Z}})$ which is fixed and u _j be an orthonormal basis of even Maass forms for SL $(2, {\mathbb{Z}})$, we prove an asymptotic formula for the average of the product of the Rankin–Selberg L-function of f and u _j and the L-function of u _j at the central value 1/2. This implies simultaneous nonvanishing results of these L-functions at 1/2.