This work studies the local theory for certain Rankin-Selberg convolutions for the standard $L$-function of degree $21n$ of generic representations of $textnormal{SO}_{2ell +1}(F)times textnormal{GL}_n(F)$ over a local field $F$. The local integrals converge in a half-plane and continue meromorphically to the whole plane. One main result is the existence of local gamma and $L$-factors. The gamma factor is obtained as a proportionality factor of a functional equation satisfied by the local integrals. In addition, Soudry establishes the multiplicativity of the gamma factor ($1