Derivatives and Exceptional Poles of the Local Exterior Square $L$-Function for $GL_m$

Let $\pi$ be an irreducible admissible representation of $GL_m(F)$, where $F$ is a non-archimedean local field of characteristic zero. We follow the method developed by Cogdell and Piatetski-Shapiro to complete the computation of the local exterior square $L$-function $L(s,\pi,\wedge^2)$ in terms of $L$-functions of supercuspidal representations via an integral representation established by Jacquet and Shalika in $1990$. We analyze the local exterior square $L$-functions via exceptional poles and Bernstein and Zelevinsky derivatives. With this result, we show the equality of the local analytic $L$-functions $L(s,\pi,\wedge^2)$ via integral integral representations for the irreducible admissible representation $\pi$ for $GL_m(F)$ and the local arithmetic $L$-functions $L(s, \wedge^2(\phi(\pi)))$ of its Langlands parameter $\phi(\pi)$ via local Langlands correspondence.