  Let $f$ be a Maass form for $SL(3, \mathbb{Z})$ which is fixed and $u_j$ be
an orthonormal basis of even Maass forms for $SL(2, \mathbb{Z}),$ we prove an
asymptotic formula for the average of the product of the Rankin-Selberg
$L$-function of $f$ and $u_j$ and the $L$-function of $u_j$ at the central
value 1/2. This implies simultaneous nonvanishing results of these
$L$-functions at $1/2.$
