On the Hurwitz Zeta Function of Imaginary Second Argument

In this work we exploit Jonqui`{e}re’s formula relating the Hurwitz zeta function to a linear combination of polylogarithmic functions in order to evaluate the real and imaginary part of ${\zeta }_H(s,ia)$ and its first derivative with respect to the first argument $s$. In particular, we obtain expressions for the real and imaginary party of ${\zeta }_H(s,ia)$ and its derivative for $s=m$ with $m\in \mathbb{Z}\mathrm{\backslash }\{1\}$ involving simpler transcendental functions.