  Let F be a genus two Siegel newform and g a classical newform, both of
squarefree levels and of equal weight l. We prove a pullback formula for
certain Eisenstein series -- thus generalizing a construction of Shimura -- and
use this to derive an explicit integral representation for the degree eight
L-function L(s, F X g). This integral representation involves the pullback of a
simple Siegel-type Eisenstein series on the unitary group GU(3,3). As an
application, we prove a reciprocity law -- predicted by Deligne's conjecture --
for the critical special values L(m, F X g) where m is an integer, 2 <= m <=
l/2-1.
