Explicit mean value theorems for toric periods and automorphic L-functions

Let F be a number field and D a quaternion algebra over F. Take a cuspidal automorphic representation π of D_𝔸^× with trivial central character and a cusp form ϕ in π. Using the prehomogeneous zeta function, we find an explicit mean value of the toric periods of ϕ with respect to quadratic algebras over F. The result can also be written as a mean value formula for the central values of automorphic L-functions twisted by quadratic characters.