Given two unitary automorphic cuspidal representationsπandπ′defined onGLm(𝔸E)andGLm'(𝔸F), respectively, withEandFbeing Galois extensions ofℚ, we consider two generalized Rankin-SelbergL-functions obtained by forcefully factoringL(s,π) and L(s,π'). We prove the absolute convergence of theseL-functions forRe(s)>1. The main difficulty in our case is that the two extension fields may be completely unrelated, so we are forced to work either “downstairs” in some intermediate extension betweenE∩Fandℚ, or “upstairs” in some extension field containing the composite extensionEF. We close by investigating some special cases when analytic continuation is possible and show that when the degrees of the extension fieldsEandFare relatively prime, the two different definitions give the same generating function.