The identities $L_2(\frac{\sqrt{5}-1}{2}) = \frac{\pi^2}{10}$ and $L_2(\frac{3-\sqrt{5}}{2}) = \frac{\pi^2}{15}$ are due to J. Landen. The rest of the identities you wrote, I suppose, could be obtained by using some other ones, like $L_2(1) = \frac{\pi^2}{6}$, $L_2(-1) = -\frac{\pi^2}{12}$, $L_2(\frac{1}{2}) = \frac{\pi^2}{12}$, due to L. Euler, and the so-called five-term relation for the Wigner-Bloch dilogarithm function, which uses an analytic continuation of the dilogarithm defined by the known infinite series. This Wigner-Bloch's dilogarithm function combines the dilogarithm (the imaginary part of its analytic continuations, to be more precise) and the usual logarithm. I think that the paper by [A. Kirillov][1] could serve you as a good reference. 

  [1]: http://arxiv.org/pdf/hep-th/9408113v2.pdf