Closed form for $\rm{Li }_2\left( -{\frac {i\sqrt {3}}{3}} \right)$

Question

In my personal research with Maple i find this closed form :

$$\operatorname{Li }_2\left( -{\frac {i\sqrt {3}}{3}} \right)={\frac {{\pi}^{2}}{24}}+{\frac {\ln \left( 2 \right) \ln \left( 3 \right) }{2}}-{\frac { \left( \ln \left( 3 \right) \right) ^{2}}{8} }+{\frac {{\operatorname{Li}_2} \left( 3 \right) }{2}}+i \left( {\frac {\pi\, \ln \left( 3 \right) }{12}}+{\frac {5\,{\pi}^{2}\sqrt {3}}{54}}-{ \frac {5\,\sqrt {3}\Psi \left( 1,1/3 \right) }{36}} \right) $$ where $\Psi \left( 1,{\frac{1}{3}} \right)$ is the trigamma function at 1/3.

But i can't remember how i find this expression.

Can someone prove this formula please ? Thanks.

Answer 1

I think it's proved in Some Nontrivial Two-Term Dilogarithm Identities of Campbell, J. M