Formulas for central critical values of twisted L-functions attached to paramodular forms

摘要

In the 1980s B\"ocherer formulated a conjecture relating the central valuesof the imaginary quadratic twists of the spin L-function attached to a Siegelmodular form $F$ to the Fourier coefficients of $F$. This conjecture has beenproved when $F$ is a lift. More recently, we formulated an analogous conjecturefor paramodular forms $F$ of prime level, even weight and in the plus-space. In this paper, we examine generalizations of this conjecture. In particular,our new formulations relax the conditions on $F$ and allow for twists by realcharacters. Moreover, these formulation are more explicit than the earlierversion. We prove the conjecture in the case of lifts and provide numericalevidence in the case of nonlifts.