Relationship between Li 2(-1/x),Li 2(-x),Li 2(-1), and log^2(x)

Theorem

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

References

1981: Leonard Lewin: Polylogarithms and Associated Functions (2nd ed.) ... (previous) ... (next): $(1.7)$