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...