A Triple Product Calculation for Gl2(r)

Let F be a number field and A = AF the ring of adeles. Let T be the subgroup of GL2 consisting of diagonal matrices with Z ⊆ T the center. Let N ⊆ GL2 be the subgroup of upper triangle unipotent matrices so that P = TN the standard Borel. Given automorphic representations π1, π2, π3 of GL2 over F such that the product of the central characters is trivial, one can consider the so-called triple product L-function L(s,Π) attached to Π = π1⊗π2⊗π3, or the completed L-function Λ(s,Π). This L-function is closely related to periods of the form