avoids the use of the incomplete gamma function, but this integral fails for "z" on the positive real axis if Re("s") ≤ 0. This expression is found by writing 2"s" Li"s"(−"z") / (−"z") = Φ("z"2, "s", 1⁄2) − "z" Φ("z"2, "s", 1), where Φ is the Lerch transcendent, and applying the Abel–Plana formula to the first Φ series and a complementary formula that involves 1 / ("e"2"πt" + 1) in place of 1 / ("e"2"πt" − 1) to the second Φ series.