  We provide a definition for an extended system of $\gamma$-factors for
products of generic representations $\tau$ and $\pi$ of split classical groups
or general linear groups over a non-archimedean local field of characteristic
$p$. We prove that our $\gamma$-factors satisfy a list of axioms (under the
assumption $p \neq 2$ when both groups are classical groups) and show their
uniqueness (in general). This allows us to define extended local $L$-functions
and root numbers. We then obtain automorphic $L$-functions $L(s,\tau \times
\pi)$, where $\tau$ and $\pi$ are globally generic cuspidal automorphic
representations of split classical groups or general linear groups over a
global function field. In addition to rationality and the functional equation,
we prove that our automorphic $L$-functions satisfy the Riemann Hypothesis.
