  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.
