I am trying to calculate some quantities related to the dynamics of acoustic vibrations in a spherical medium, and I have come across the following series: $$ S = \sum_{n=1}^\infty \frac{e^{iz_{2,n}\Omega}}{z_{2,n}}. $$ where the numbers $z_{2,n}$ are the zeros of the spherical Bessel function of order 2 ($j_2(z)$), or equivalently the zeros of the regular Bessel function $J_{5/2}(z)$, excluding $z=0$. $\Omega$ stands for some real parameter.
Noting that for $z$ large enough one can approximate $z_{2,n} \approx (n+1)\pi$, I have been able to approximate $$ S \approx \sum_{n=1}^\infty \frac{e^{i\Omega\pi(n+1)}}{\pi(n+1)} = -\frac{e^{i\pi\Omega} +Log\left(1-e^{i\pi\Omega}\right)}{\pi}. $$ However, I am not very satisfied with this result as I think its accuracy is quite important in my case. Does anyone here know whether the sum $S$ has some exact solution, or how can I get a more accurate result?
Thanks in advance!