I have the expression
$$\tag{1} \operatorname{Li}_{1/2}(z)-\operatorname{Li}_{1/2}(z^*) $$
Where $\operatorname{Li}$ is the polylogarithm and $^*$ denotes complex conjugation. The expression is similar to
$$ \operatorname{Li}_{s}(e^{2\pi ia})+e^{i\pi s}\operatorname{Li}_{s}(e^{-2\pi ia})=\frac{(2\pi)^s}{\Gamma(s)}e^{i\pi s/2}\zeta(1-s,a) $$
from DLMF, where $\zeta$ is the Hurwitz zeta function. This makes me wonder if a similar compact representation of (1) exists. The duplication and inversion formulas are also almost applicable, but not quite (or, so it seems to me).
For context, (1) results from performing the sum
$$ \sum\limits_{n=1}^\infty n^{-1/2}e^{-an}\sin(bn) $$
