  Let $F$ be a number field and $D$ a quaternion algebra over $F$. Take a
cuspidal automorphic representation $\pi$ of $D_{\mathbb{A}}^\times$ with
trivial central character and a cusp form $\phi$ in $\pi$. Using the
prehomogeneous zeta function, we find an explicit mean value of the toric
periods of $\phi$ 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.
