Renormalization of Certain Integrals Defining Triple Product L-functions

We obtain special values results for the triple product Lfunction attached to a Hilbert modular cuspidal eigenform over a totally real quadratic number field and an elliptic modular cuspidal eigenform, both of level one and even weight. Replacing the elliptic modular cusp form by a specified Eisenstein series, we renormalize the integral defining the triple product L-function in order to obtain an integral representation for a product of Asai L-functions. We hope in further work to extend these results to triple-product L-functions attached to automorphic representations and then study the critical values of this renormalized triple product.