  Let $f,g,h$ be three normalized cusp newforms of weight $2k$ on $\Gamma_0(N)$
which are eigenforms of Hecke operators. We use Ichino's period formula
combined with a relative trace formula to show exact averages of
$L(3k-1,f\times g\times h)$. We also present some applications of the average
formulas to the nonvanishing problem, giving a lower bound on the number of
nonvanishing central $L$-values when one of the forms is fixed.
