L-Functions for Holomorphic Forms on GSp(4) x GL(2) and Their Special Values

We provide an explicit integral representation for L-functions of pairs (F, g), where F is a holomorphic genus two Siegel newform and g a holomorphic elliptic newform, both of square-free levels and of equal weights. When F, g have level one, this was earlier known by the work of Furusawa. The extension is not straightforward. Our methods involve precise double-coset and volume computations as well as an explicit formula for the Bessel model for GSp(4) in the Steinberg case; the latter is possibly of independent interest. As an application, we prove an algebraicity result for a critical value of L(s, F x g). This is in the spirit of known results on critical values of triple product L-functions, also of degree eight, though there are significant differences