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  "body": "\n## Summary\n\nThe Lerch zeta function is the series $\\sum_{n \\ge 0} e^{2\\pi i \\lambda n} (n + \\alpha)^{-s}$, a generalized [[Dirichlet series|8MWTEF6462]] with a phase factor and a shift.\nIn the closely related form $\\Phi(z, s, a) = \\sum_{n \\ge 0} z^n (n + a)^{-s}$ it is called the Lerch transcendent.\nIt contains the [[Hurwitz zeta function|G4SW7HJG88]], the [[Riemann zeta function|1XWBY4N0VT]], and the [[polylogarithm|96CHDZKGQ0]] as special cases.\nDirichlet $L$-functions are finite linear combinations of its values.\nSo identities proved for it hold for all of them at once.\n\n## Explanations\n\n### L1 human\n\nThe Riemann zeta function adds up $1/n^s$.\nThe Hurwitz zeta function shifts the integers to $n + \\alpha$, and the polylogarithm weights the terms by powers $z^n$.\nThe Lerch zeta function does both: it adds up $z^n / (n + \\alpha)^s$.\nFor instance, with $z = -1$, $\\alpha = 1$ and $s = 1$, the sum is $1 - \\frac12 + \\frac13 - \\cdots = \\log 2$.\n",
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