A Note on Colored Tornheim's Double Series

In this short note, we provide an explicit formula to compute every colored double Tornheim's series by using double polylogarithm values at roots of unity. When the colors are given by $\pm 1$ our formula is different from that of Tsumura [On alternating analogues of Tornheim's double series II, Ramanujan J. 18 (2009), 81-90] even though numerical data confirm both are correct in almost all the cases. This agreement can also be checked rigorously by using regularized double shuffle relations of the alternating double zeta values in weights less than eight.

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