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Fix $g$ a Hecke-Maass form for $SL_3(\mathbb{Z})$. Let $q$ be any large primenumber. In the family of holomorphic newforms $f$ of level $q$ and fixedweight, we find the average value of the product $L(\half,g\timesf)L(\half,f)$. From this we derive a result on the simultaneous non-vanishingof these $L$-functions at the central point.

Simultaneous non-vanishing of $GL(3)\times GL(2)$ and GL(2) $L$-functions