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The Rankin-Selberg method for studying Langlands' automorphic $L$-functionsis to find integral representations, involving certain Fourier coefficients ofcusp forms and Eisenstein series, for these functions. In this thesis wedevelop the local theory for generic representations of special orthogonalgroups. 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 functionalequation, which is used to define a $\gamma$-factor. We show that, as expected,the $\gamma$-factor is identical with Shahidi's $\gamma$-factor. The analyticproperties of the integrals are condensed into a notion of a greatest commondivisor (g.c.d.). We establish certain bounds on the g.c.d. and relate it tothe $L$-function defined by Shahidi in several cases, thereby providing anotherpoint of view on the $L$-function, linking it to the poles of the integrals. Inparticular, in the tempered case under a reasonable assumption the g.c.d. isequal to the $L$-function. Finally, this study includes the computation of theintegrals with unramified data. This work may lead to new applications of thedescent method, as well as aid in analyzing the poles of the global$L$-function.

On the Local Theory of Rankin-Selberg Convolutions for $\mathrm{SO_{2l}\times GL_{n}}$