TITLE: Simplification of a trilogarithm of a complex argument
QUESTION [14 upvotes]: Is it possible to simplify the following expression?
$$\large\Im\,\operatorname{Li}_3\left(-e^{\xi\,\left(\sqrt3-\sqrt{-1}\right)-\frac{\pi^2}{12\,\xi}\left(\sqrt3+\sqrt{-1}\right)}\right)$$
where
$$\large\xi=\frac{\sqrt[3]3}6\sqrt[3]{27+\sqrt3\,\sqrt{243-\pi^6}}$$
and $\Im\,\operatorname{Li}_3(z)$ denotes the imaginary part of the trilogarithm.

REPLY [13 votes]: Step 1: Note that the trilogarithm argument
$$z=e^{2\pi i x}=-e^{\xi\,\left(\sqrt3-\sqrt{-1}\right)-\frac{\pi^2}{12\,\xi}\left(\sqrt3+\sqrt{-1}\right)}$$
lies on the unit circle.
Step 2: Use that
$$\operatorname{Li}_n(e^{2\pi i x})+(-1)^n\operatorname{Li}_n(e^{-2\pi i x})=-\frac{(2\pi i)^n}{n!}B_n(x),$$
where $B_n(x)$ is the $n$th Bernoulli polynomial. Of course, here $n=3$.
Step 3: Simplify. 

I am pretty sure this will give you the answer $\frac12$ of @VladimirReshetnikov .