  The Rankin-Selberg method for studying Langlands' automorphic $L$-functions
is to find integral representations, involving certain Fourier coefficients of
cusp forms and Eisenstein series, for these functions. In this thesis we
develop the local theory for generic representations of special orthogonal
groups. We study the local integrals for $SO_{2l}\times GL_{n}$, where
$SO_{2l}$ is the special even orthogonal group, either split or quasi-split,
over a local non-Archimedean field. These integrals admit a functional
equation, which is used to define a $\gamma$-factor. We show that, as expected,
the $\gamma$-factor is identical with Shahidi's $\gamma$-factor. The analytic
properties of the integrals are condensed into a notion of a greatest common
divisor (g.c.d.). We establish certain bounds on the g.c.d. and relate it to
the $L$-function defined by Shahidi in several cases, thereby providing another
point of view on the $L$-function, linking it to the poles of the integrals. In
particular, in the tempered case under a reasonable assumption the g.c.d. is
equal to the $L$-function. Finally, this study includes the computation of the
integrals with unramified data. This work may lead to new applications of the
descent method, as well as aid in analyzing the poles of the global
$L$-function.
