OFFSET
1,1
COMMENTS
Nesterenko proves that this constant is transcendental (he cites Chudnovsky as the first to show this); in fact it is algebraically independent of Pi and e^Pi over Q. - Charles R Greathouse IV, Nov 11 2013
REFERENCES
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 33.
LINKS
Harry J. Smith, Table of n, a(n) for n = 1..1000
William Duke and Özlem Imamoḡlu, Special values of multiple gamma functions, Journal de théorie des nombres de Bordeaux, Vol. 18. No. 1 (2006), pp. 113-123.
Greg J. Fee and Simon Plouffe, Gamma(1/4) to 25000 digits.
Yu. V. Nesterenko, Modular functions and transcendence questions, Sbornik: Mathematics, Vol. 187, No. 9 (1996), pp. 1319-1348. (English translation)
Simon Plouffe, Gamma(1/4) to 250000 digits.
Dan Romik, On Viazovska's modular form inequalities, arXiv:2303.13427 [math.NT], 2023.
Eric Weisstein's World of Mathematics, Gamma Function.
FORMULA
From Amiram Eldar, Jun 12 2021: (Start)
Equals sqrt(2*sqrt(2*Pi^3)*G), where G is Gauss's constant (A014549).
Equals (2*Pi)^(3/4) * Product_{k>=1} tanh(k*Pi/2) (Duke and Imamoḡlu, 2006). (End)
Equals Product_{n>=1} exp((2*(6*n + 1)*(1 - beta(n)) - (eta(n) - 1))/(4*n)), where eta(n) and beta(n) are the Dirichlet eta and beta functions, respectively. - Antonio Graciá Llorente, Sep 05 2024
EXAMPLE
3.6256099082219083119306851558676720029951676828800654674333...
MAPLE
evalf(GAMMA(1/4));
MATHEMATICA
RealDigits[Gamma[1/4], 10, 110][[1]] (* Bruno Berselli, Dec 13 2012 *)
PROG
(PARI) default(realprecision, 1080); x=gamma(1/4); for (n=1, 1000, d=floor(x); x=(x-d)*10; write("b068466.txt", n, " ", d)); \\ Harry J. Smith, Apr 19 2009
(Magma) R:= RealField(100); SetDefaultRealField(R); Gamma(1/4); // G. C. Greubel, Mar 10 2018
CROSSREFS
KEYWORD
AUTHOR
Benoit Cloitre, Mar 10 2002
STATUS
approved