Integral representation for L-functions for GSp4×GL2

Let π be a cuspidal, automorphic representation of GSp4 attached to a Siegel modular form of degree 2. We refine the method of Furusawa [M. Furusawa, On L-functions for GSp(4)×GL(2) and their special values, J. Reine Angew. Math. 438 (1993) 187–218] to obtain an integral representation for the degree-8 L-function L(s,π×τ), where τ runs through certain cuspidal, automorphic representation of GL2. Our calculations include the case of any representation with unramified central character for the p-adic components of τ, and a wide class of archimedean types including Maaß forms. As an application we obtain a special value result for L(s,π×τ).