cp's OEIS Frontend

A307065 Decimal expansion of the negative real attracting fixed point of Э(s) = (1 - 2^s) * (1 - 2^(1 - s)) * gamma(s) * zeta(s) * beta(s) / Pi^s.

%I A307065 #17 Sep 24 2021 20:41:58
%S A307065 1,7,8,4,8,3,0,9,7,1,4,2,9,5,4,5,7,0,2,8,6,0,5,7,5,4,6,6,4,2,0,3,7,0,
%T A307065 7,6,9,9,7,8,3,1,5,9,1,5,5,9,5,0,7,2,6,1,0,4,4,7,8,5,7,2,1,3,8,6,4,9,
%U A307065 3,3,1,7,9,2,4,1,3,6,1,7,4,9,5,3,4,0,3,7,1,7,8,9,9,8,8,7,1,2,1,7
%N A307065 Decimal expansion of the negative real attracting fixed point of Э(s) = (1 - 2^s) * (1 - 2^(1 - s)) * gamma(s) * zeta(s) * beta(s) / Pi^s.
%C A307065 Ossicini's function Э(s) is constructed to remove the poles of gamma(s) and zeta(s) along with the trivial zeros of zeta(s) and (Dirichlet) beta(s).  Its zeros include the nontrivial zeros of zeta(s) and beta(s), and complex zeros contributed by (1 - 2^s) and (1 - 2^(1 - s)) at regular intervals of 2*Pi/log(2) on the lines Re(s) = {0, 1}.
%D A307065 A. Ossicini, An alternative form of the functional equation for Riemann's Zeta function, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 56 (2008/09), 95-111.
%H A307065 Andrea Ossicini, <a href="https://arxiv.org/abs/1206.4494">An Alternative Form of the Functional Equation for Riemann's Zeta Function, II</a>, arXiv:1206.4494 [math.HO], 2012-2014.
%e A307065 -0.1784830971429545702860575466420370769978315915595...
%t A307065 f[s_] := s - (1 - 2^s)(1 - 2^(1-s)) Gamma[s] Zeta[s] ((HurwitzZeta[s, 1/4] - HurwitzZeta[s, 3/4])/(4 Pi)^s);
%t A307065 s0 = s /. FindRoot[f[s], {s, -1/5}, WorkingPrecision -> 100];
%t A307065 RealDigits[s0][[1]] (* _Jean-François Alcover_, May 07 2019 *)
%o A307065 (PARI) solve(s = -1/2, -1/8, s - (1 - 2^s) * (1 - 2^(1 - s)) * gamma(s) * zeta(s) * (zetahurwitz(s, 1/4) - zetahurwitz(s, 3/4)) / (4 * Pi)^s)
%Y A307065 Cf. A069857, A069995.
%K A307065 nonn,cons,easy
%O A307065 0,2
%A A307065 _Reikku Kulon_, Mar 22 2019