  We study the sum $\ds\zeta_H(s)=\sum_j E_j^{-s}$ over the eigenvalues $E_j$
of the Schrdinger equation in a spherical domain with Dirichlet walls, threaded
by a line of magnetic flux. Rather than using Green's function techniques, we
tackle the mathematically nontrivial problem of finding exact sum rules for the
zeros of Bessel functions $J_{\nu}$, which are extremely helpful when seeking
numerical approximations to ground state energies. These results are
particularly valuable if $\nu$ is neither an integer nor half an odd one.
