  We prove a conjecture of B. Gross and D. Prasad about determination of
generic $L$-packets in terms of the analytic properties of the adjoint
$L$-function for $p$-adic general even spin groups of semi-simple ranks 2 and
3. We also explicitly write the adjoint $L$-function for each $L$-packet in
terms of the local Langlands $L$-functions for the general linear groups.
