On the central critical value of the triple product $L$-function
Siegfried Boecherer and Rainer Schulze-Pillot
We compute the central critical value of the triple product $L$-function associated to three cusp forms $f_1,f_2,f_3$ with trivial character for groups $\Gamma_0(N_i)$ with square free levels $N_i$ not all of which are $1$ and weights $k_i$ satisfying $k_1\ge k_2\ge k_3$ and $k_1 \lt k_2 + k_3$. This generalizes work of Gross and Kudla and gives an alternative classical proof of their results in the case $N_1=N_2=N_3$ with $k_1=k_2=k_3=2$.