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A203138
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Decimal expansion of Gamma(1/24).
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9
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2, 3, 4, 6, 2, 4, 8, 7, 6, 9, 3, 1, 8, 3, 3, 1, 9, 8, 8, 1, 3, 8, 5, 7, 1, 1, 4, 6, 9, 5, 8, 6, 2, 9, 4, 9, 3, 0, 4, 3, 3, 3, 6, 5, 1, 3, 4, 0, 0, 4, 6, 1, 0, 1, 6, 4, 7, 3, 9, 7, 9, 8, 4, 7, 5, 8, 2, 5, 1, 5, 0, 1, 1, 4, 0, 1, 8, 3, 9, 7, 7, 6, 9, 4, 3, 4, 9, 9, 1, 7, 4, 6, 5, 9, 4, 9, 5, 9, 7
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OFFSET
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2,1
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COMMENTS
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In the article by Vidunas, the third formula on page 13 is wrong. The exponent of the term K((sqrt(3)-1)/(2*sqrt(2)))^(1/3) is wrong. It should be Gamma(1/24) = Pi^(1/24) * 2^(89/36) * 3^(25/48) * sqrt(1+sqrt(2)) * (sqrt(3)-1)^(1/4) * K(1/sqrt(2))^(1/4) * K((sqrt(3)-1)/(2*sqrt(2)))^(1/6) * K((2-sqrt(3))*(sqrt(3)-sqrt(2)))^(1/2). - Vaclav Kotesovec, Apr 21 2024
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LINKS
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FORMULA
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Equals 2^(13/12) * 3^(9/16) * Pi^(1/4) * (sqrt(3) - 1)^(1/4) * sqrt((1 + sqrt(2)) * Gamma(1/3) * Gamma(1/4)) * EllipticTheta(3, 0, exp(-Pi*sqrt(6))).
Equals 2^(35/24) * 3^(3/8) * sqrt(Pi*(1 + sqrt(2)) * Gamma(1/12) / (1 + sqrt(3))) * EllipticTheta(3, 0, exp(-Pi*sqrt(6))). (End)
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EXAMPLE
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23.462487693183319881385711469586294930433365134004610164739...
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MAPLE
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evalf(Pi^(1/24) * 2^(89/36) * 3^(25/48) * sqrt(1+sqrt(2)) * (sqrt(3)-1)^(1/4) * EllipticK(1/sqrt(2))^(1/4) * EllipticK((sqrt(3)-1)/(2*sqrt(2)))^(1/6) * EllipticK((2-sqrt(3))*(sqrt(3)-sqrt(2)))^(1/2), 110); # Vaclav Kotesovec, Apr 21 2024
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MATHEMATICA
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RealDigits[Gamma[1/24], 10, 100][[1]] (* G. C. Greubel, Mar 10 2018 *)
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PROG
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(PARI) default(realprecision, 100); gamma(1/24) \\ G. C. Greubel, Mar 10 2018
(Magma) SetDefaultRealField(RealField(100)); Gamma(1/24); // G. C. Greubel, Mar 10 2018
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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