Relationship between Li 2(1),Li 2(-1), and pi

Theorem

The following formula holds: $$2\mathrm{Li}_2(1) = 2\mathrm{Li}_2(-1) + \dfrac{\pi^2}{2},$$ where $\mathrm{Li}_2$ denotes the dilogarithm and $\pi$ denotes pi.

References

1981: Leonard Lewin: Polylogarithms and Associated Functions (2nd ed.) ... (previous) ... (next): $(1.8)$