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  "body": "\n## Example\n\nHere are some values of the [[gamma function|0V3FZMVB4K]].\n\n| $z$ | $\\Gamma(z)$ |\n|---|---|\n| $n \\in \\set{1, 2, 3, \\ldots}$ | $(n - 1)!$ |\n| $\\tfrac12$ | $\\sqrt{\\pi}$ |\n| $\\tfrac32$ | $\\tfrac12\\sqrt{\\pi}$ |\n| $\\tfrac52$ | $\\tfrac34\\sqrt{\\pi}$ |\n| $n + \\tfrac12$, $n \\ge 0$ | $\\dfrac{(2n)!}{4^n \\, n!}\\sqrt{\\pi}$ |\n| $-\\tfrac12$ | $-2\\sqrt{\\pi}$ |\n| $-\\tfrac32$ | $\\tfrac43\\sqrt{\\pi}$ |\n\nChecks.\nThe first row is [[the gamma function extends the factorial|0FDCW77H44]].\nThe value at $\\tfrac12$ is [[Gamma of one half|WM5FNBB7K0]].\nThe [[functional equation|ZCB806EV8A]] $\\Gamma(z + 1) = z\\Gamma(z)$ gives the others: $\\Gamma(\\tfrac52) = \\tfrac32 \\cdot \\tfrac12 \\sqrt{\\pi} = \\tfrac34\\sqrt\\pi$, and $\\Gamma(-\\tfrac12) = \\Gamma(\\tfrac12)/(-\\tfrac12) = -2\\sqrt\\pi$, and $\\Gamma(-\\tfrac32) = \\Gamma(-\\tfrac12)/(-\\tfrac32) = \\tfrac43\\sqrt\\pi$.\nFor $n = 2$ the general half-integer formula gives $\\frac{24}{16 \\cdot 2}\\sqrt{\\pi} = \\tfrac34\\sqrt{\\pi}$, in agreement.\n\nValues at other rational points are less explicit, but the [[reflection formula|WF7KQAQTS2]] gives products:\n$$\\Gamma\\!\\left(\\tfrac13\\right)\\Gamma\\!\\left(\\tfrac23\\right) = \\frac{\\pi}{\\sin(\\pi/3)} = \\frac{2\\pi}{\\sqrt3}, \\qquad \\Gamma\\!\\left(\\tfrac14\\right)\\Gamma\\!\\left(\\tfrac34\\right) = \\frac{\\pi}{\\sin(\\pi/4)} = \\pi\\sqrt2.$$\n\n## Remarks\n\nThe values $\\Gamma(\\tfrac13)$ and $\\Gamma(\\tfrac14)$ are known to be [[transcendental numbers|0VTQXTCR2G]].\nThe value $\\Gamma(\\tfrac14)$ appears in the arc length of the lemniscate and in elliptic integrals.\n\n## Explanations\n\n### L1 human\n\nAt whole numbers the gamma function gives factorials, shifted by one.\nAt halves it gives rational multiples of $\\sqrt{\\pi}$, and the sign alternates for negative halves.\nAt thirds and quarters the single values are new constants, but some of their products are simple.\n",
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