With Maple i find this closed form:
${\it {Li_2}} \left( 1-{\frac {\sqrt {2}}{2}}-i \left( 1-{\frac {\sqrt { 2}}{2}} \right) \right)$=$-{\frac {{\pi}^{2}}{64}}-{\frac { \left( \ln \left( 1+\sqrt {2} \right) \right) ^{2}}{2}}-{\frac {{\it {Li_2}} \left( -1-\sqrt {2} \right) }{2}}+{\frac {i}{32}} \left( 4\,\pi\,\ln \left( 1+\sqrt {2} \right) -24\,G \right)$
where $G$ is the Catalan's constant and $Li_2$ is the dilogarithm function or Spence's function.
If you have got any ideas please for proving this identity.
