  We prove three main results: all Langlands-Shahidi automorphic $L$-functions
over function fields are rational; after twists by highly ramified characters
our automorphic $L$-functions become polynomials; and, if $\pi$ is a globally
generic cuspidal automorphic representation of a split classical group or a
unitary group ${\bf G}_n$ and $\tau$ a cuspidal (unitary) automorphic
representation of a general linear group, then $L(s,\pi \times \tau)$ is
holomorphic for $\Re(s) > 1$ and has at most a simple pole at $s=1$. We also
prove the holomorphy and non-vanishing of automorphic exterior square,
symmetric square and Asai L-functions for $\Re(s) > 1$. Finally, we complete
previous results on functoriality for the classical groups over function
fields.
