On L-functions for Quaternion Unitary Groups of Degree 2 and GL(2) (with an Appendix by M. Furusawa and A. Ichino)

We consider a certain Rankin-Selberg integral which represents the tensor product L-function for GL(2) and a similitude quaternion unitary group of degree 2, which is an inner form of GSp(4). As a consequence, when the base field is Q and the similitude quaternion unitary group is isomorphic to GSp(4) over R, we prove an algebraicity result for special values of the L-function, which is a generalization of the previous results for GSp(4) x GL(2). Assuming the existence of the transfer from similitude quaternion unitary groups to GSp(4), we obtain a certain period relation as a corollary of the algebraicity result.