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 - the case of p = 2 included. 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 G. Henniart and the author allows us to show that the definitions provided in this article are in accordance with the local Langlands conjecture for 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.