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  "body": "\n## Statement\n\nLet $\\Gamma$ be the [[gamma function|0V3FZMVB4K]], extended to a [[meromorphic function|0NSD1TPHR0]] on $\\C$.\nFor every $z \\in \\C \\setminus \\set{0, -1, -2, \\ldots}$, $\\Gamma(z) \\ne 0$.\n\n## Remarks\n\nFor $z \\notin \\Z$, the [[reflection formula|WF7KQAQTS2]] gives $\\Gamma(z)\\Gamma(1 - z) = \\pi/\\sin(\\pi z)$, and the right side is finite and nonzero, so $\\Gamma(z) \\ne 0$.\nFor a positive integer $n$, $\\Gamma(n) = (n-1)! \\ne 0$.\nThe points $0, -1, -2, \\ldots$ are excluded because $\\Gamma$ has poles there, not values.\nA consequence is that the [[reciprocal gamma function|PY6K9AWR33]] $1/\\Gamma$ is entire, with zeros exactly at $0, -1, -2, \\ldots$.\nIn Mathlib the statement is `Complex.Gamma_ne_zero`.\n\n## Explanations\n\n### L1 human\n\nThe gamma function never takes the value zero.\nIt can become very large near its poles and very small for large negative arguments, but it never vanishes.\n\n### L2 human\n\nBecause $\\Gamma$ has no zeros, its logarithmic derivative, the [[digamma function|HWRRXB0YXS]] $\\psi = \\Gamma'/\\Gamma$, is meromorphic with poles only at $0, -1, -2, \\ldots$.\nIt also allows a holomorphic logarithm of $\\Gamma$ on the slit plane, the [[log-gamma function|JT95B44G4G]].\n",
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