Holomorphy of Rankin tripleL-functions; special values and root numbers for symmetric cubeL-functions

In this paper we prove the holomorphy of Rankin tripleL-functions for three cusp forms on GL(2) on the entire complex plane, if at least one of them is non-monomial. We conclude the paper by proving the equality of our root numbers for the third and the fourth symmetric powerL-functions with those of Artin through the local Langlands correspondence. We also revisit Deligne’s conjecture on special values of symmetric cubeL-functions. Partially supported by NSF grant DMS9610387. Partially supported by NSF grant DMS9970156.