Central Values of Degree Six L-functions

Let κ′, κ ≥ 3 be two odd integers. Let f (resp. g) be a normalized holomorphic modular form of weight 2κ (resp. κ′ + 1) and level one on the upper half plane h. Assume that they are Hecke eigenforms. Let L(s, Sym g×f) be the completed degree six L-function and we normalize so that s = 1 2 is the center of symmetry. Let 〈−,−〉 be the Petersson inner product, defined using the usual measure on h so that the volume of Γ0(1)\h equals π3 . Let c(f) be the fundamental period of f defined in [Shi77]. The goal of this short note is to prove the following result.