Let p be the automorphic representation of GSp4 (A) generated by a full level cuspidal Siegel eigenform that is not a Saito-Kurokawa lift, and t be an arbitrary cuspidal, automorphic representation of GL? (A). Using Furusawa's integral representation for GSp? X GL? combined with a pullback formula involving the unitary group GU (3,3), the authors prove that the L-functions L(s,p X t are ``nice”.

The converse theorem of Cogdell and Piatetski-Shapiro then implies that such representations p have a functorial lifting to a cuspidal representation of GL? (A). Combined with the exterior-square lifting of Kim, this also leads to a functorial lifting of p to a cuspidal representation of GL5 (A).

As an application, the authors obtain analytic properties of various L-functions related to full level Siegel cusp forms. They also obtain special value results for GSp? X GL? and GSp4 X GL?.