  We provide an explicit integral representation for L-functions of pairs (F,g)
where F is a holomorphic genus 2 Siegel newform and g a holomorphic elliptic
newform, both of squarefree 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. We apply
our integral representation to prove an algebraicity result for a critical
special value of L(s, F \times g). This is in the spirit of known results on
critical values of triple product L-functions, also of degree 8, though there
are significant differences.
