%I #55 Sep 28 2022 13:56:31

%S 3,9,4,7,8,4,1,7,6,0,4,3,5,7,4,3,4,4,7,5,3,3,7,9,6,3,9,9,9,5,0,4,6,0,

%T 4,5,4,1,2,5,4,7,9,7,6,2,8,9,6,3,1,6,2,5,0,5,6,5,3,3,9,7,5,0,4,8,8,0,

%U 1,7,9,2,8,9,6,7,6,8,2,0,9,7,2,0,0,7

%N Decimal expansion of (2*Pi)^2.

%C This constant appears in Kepler's 3rd Law, T^2 = (2*Pi)^2/GM*a^3 where a is the semi-major axis of a planet orbiting the Sun, T is its period, and GM is the standard gravitational parameter. - _Raphie Frank_, Dec 13 2012

%C García & Marco give a generalized zeta regularization by which this is the value of the product of the primes. - _Charles R Greathouse IV_, Jun 17 2013

%H J.-P. Allouche, <a href="https://arxiv.org/abs/1906.10532">The zeta-regularized product of odious numbers</a>, arXiv:1906.10532 [math.NT], 2019.

%H Hyperphysics, <a href="http://hyperphysics.phy-astr.gsu.edu/hbase/kepler.html">Kepler's Laws</a>

%H E. Muñoz García and R. Pérez Marco, <a href="https://webusers.imj-prg.fr/~ricardo.perez-marco/publications/articles/CMP2008.pdf">The product over all primes is 4Pi^2</a>, Communications in Mathematical Physics, Vol. 277, No. 1 (2008), pp. 69-81.

%H Hengguang Li and Jeffrey S. Ovall, <a href="http://www.hli.wayne.edu/research/publications/LO15.pdf">A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential</a>, Discrete and Continuous Dynamical Systems Series B, Volume 20, Number 5, July 2015, pp. 1377-1391. Also doi:10.3934/dcdsb.2015.20.1377

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Pendulum">Pendulum</a>.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%F Equals Product_{k=1..10, gcd(k,10)==1} Gamma(k/10) = Gamma(1/10)*Gamma(3/10)*Gamma(7/10)*Gamma(9/10). - _Amiram Eldar_, Jun 12 2021

%F Equals lim_{n->oo} |B(2*n)/B(2*n+2)|*(2*n+1)*(2*n+2), where B(n) denotes the n-th Bernoulli number. - _Peter Luschny_, Dec 09 2021

%e 39.4784176043574344753379639995046045412547976289631...

%t RealDigits[(2*Pi)^2,10,120][[1]] (* _Harvey P. Dale_, Mar 27 2019 *)

%o (PARI) 4*Pi^2 \\ _Charles R Greathouse IV_, Jun 17 2013

%Y Cf. A000796, A002388, A019692, A212003.

%K nonn,cons

%O 2,1

%A _Omar E. Pol_, Aug 11 2012