  Furusawa has given an integral representation for the degree 8 L-function of
GSp(4) x GL(2) and has carried out the unramified calculation. The local p-adic
zeta integrals were calculated in our earlier work under the assumption that
the GSp(4) representation \pi is unramified and the GL(2) representation \tau
has conductor p. In the present work we generalize to the case where the GL(2)
representation has arbitrarily high conductor. The result is that the zeta
integral represents the local Euler factor L(s,\pi \times \tau) in all cases.
As a global application we obtain a special value result for a GSp(4) x GL(2)
global L-function coming from classical holomorphic cusp forms with arbitrarily
high level for the elliptic modular form.
