RANKIN-SELBERG CONVOLUTIONS FOR SO(2L- LOCAL THEORY(1) X GL(N) )

In this Memoir we study the local theory for Rankin Selberg convolutio ns for the degree 2ln standard L-function of generic representations o f SO2l+1(F) x GL(n)(G) over a local field F. We prove the convergence in a half plane of the local integrals, their meromorphic continuation (if F is archimedean only for l greater-than-or-equal-to n) and the e xistence of local gamma and L-factor. Our main result is that of the m ultiplicativity of the gamma factor (l < n, first variable). As a cons equence of the proof of this multiplicativity we obtain the unramified computation of the local integral, when all data is unramified.