Are there any equivalent representations of the following (real valued) sum, in particular that are suitable for evaluation as $z\rightarrow0$ ? $$ S=\sum_{k=-\infty}^\infty \frac{i^k(z-2ik)}{(\rho^2+(z-2ik)^2)^{3/2}} $$

I am aware that $S$ resembles a coulomb force sum which can be rearranged using Lekner summation into a seires of Bessel functions, but the factor $i^k$ seems to prohibit this transformation.