  Let $F$ be a non-archimedean local field of odd characteristic $p > 0$. In
this paper, we consider local exterior square $L$-functions
$L(s,\pi,\wedge^2)$, Bump-Friedberg $L$-functions $L(s,\pi,BF)$, and Asai
$L$-functions $L(s,\pi,As)$ of an irreducible admissible representation $\pi$
of $GL_m(F)$. In particular, we establish that those $L$-functions, via the
theory of integral representations, are equal to their corresponding Artin
$L$-functions $L(s,\wedge^2(\phi(\pi)))$,
$L(s+1/2,\phi(\pi))L(s,\wedge^2(\phi(\pi)))$, and $L(s,As(\phi(\pi)))$ of the
associated Langlands parameter $\phi(\pi)$ under the local Langlands
correspondence. These are achieved by proving the identity for irreducible
supercuspidal representations, exploiting the local to global argument due to
Henniart and Lomeli.
