  In the 1980s B\"ocherer formulated a conjecture relating the central values
of the imaginary quadratic twists of the spin L-function attached to a Siegel
modular form $F$ to the Fourier coefficients of $F$. This conjecture has been
proved when $F$ is a lift. More recently, we formulated an analogous conjecture
for 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 real
characters. Moreover, these formulation are more explicit than the earlier
version. We prove the conjecture in the case of lifts and provide numerical
evidence in the case of nonlifts.
