$\operatorname{Re}(\operatorname{Li}_3(z))$ for real $z\geq1$ in terms of elementary functions?

Question

According to the article by Wood, D. "The Computation of Polylogarithms. Technical Report 15-92*", listed in the references about the polylogarithm on the Wikipedia, there is a form in terms of elementary functions for the imaginary part of the polylogarithm of integer order $n>0$ and real argument $z\geq1$,

\begin{equation} \operatorname{Im}(\operatorname{Li}_n(z)) = -\frac{\pi\mu^{n-1}}{\Gamma(n)},\text{with } \mu=\ln(z). \end{equation}

My question is: Is there an analogous formula expressing $\operatorname{Re}(\operatorname{Li}_3(z))$ in terms of elementary functions in the same event of real $z\geq 1$?

Thank you!

Answer 1

For real $z>1$ you can use the real part of the inversion formula with $n=3$ to get $$\Re\,\mathrm{Li}_3(z) = \Re\,\mathrm{Li}_3(z^{-1}) + \frac{1}{3}\pi^2 \ln z - \frac{1}{6}(\ln z)^3.$$

So if there would be an elementary expression for $\Re\,\mathrm{Li}_3(z)$ for real $z > 1$, there also would be an elementary expression for $0<z<1$, which is not the case (as far as I know).