Critical values of L-functions on G Sp2 x Gl2

This paper deals with Deligne's conjecture on the critical values of L-functions. Let Z(G circle times h)(s) denote the tensor product L-function attached to a Siegel modular form G of weight k and an elliptic cusp form h of weight l. We assume that the first Fourier-Jacobi coefficient of G is not identically zero. Then Deligne's conjecture is fully proven for Z(G circle times h)(s), when l <= 2k-2 and partly for the remaining case.