  We give a proof of the existence of Asai, exterior square, and symmetric
square local $L$-functions, $\gamma$-factors and root numbers in characteristic
$p$, including the case of $p = 2$. Our study is made possible by developing
the Langlands-Shahidi method over a global function field in the case of a
Siegel Levi subgroup of a split classical group or a quasi-split unitary group.
The resulting automorphic $L$-functions are shown to satisfy a rationality
property and a functional equation. A uniqueness result of the author and G.
Henniart allows us to show that the definitions provided in this article are in
accordance with the local Langlands conjecture for ${\rm GL}_n$. Furthermore,
in order to be self contained, we include a treatise of $L$-functions arising
from maximal Levi subgroups of general linear groups.
