Simplification of a trilogarithm of a complex argument

Question

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.

Answer 1

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.

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