GL(2) x GS(p2) L-values and Hecke eigenvalue congruences

We find experimental examples of congruences of Hecke eigen-values between automorphic representations of groups such as GSp(2)(A), SO(4, 3)(A) and SO(5, 4)(A), where the prime modulus should, for various reasons, appear in the algebraic part of a critical "tensor-product" L-value associated to cuspidal automorphic representations of GL(2)(A) and GS(p2)(A). Using special techniques for evaluating L-functions with few known coefficients, we compute sufficiently good approximations to detect the anticipated prime divisors.