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A387393 Decimal expansion of the imaginary part of the smallest complex solution to zeta(z) = zeta(1-z).

%I A387393 #17 Sep 03 2025 22:20:35
%S A387393 3,4,3,6,2,1,8,2,2,6,0,8,6,9,6,1,5,9,1,6,5,5,9,6,5,4,2,5,6,5,6,4,7,2,
%T A387393 8,8,8,0,8,8,5,7,8,0,8,2,9,7,5,2,0,5,3,2,6,5,3,4,1,3,9,4,3,8,8,8,0,3,
%U A387393 4,2,8,6,2,3,1,8,7,3,4,0,8,6,8,7,4,6,3,1,1,7,6,6,0,3,9,4,3,7,2,8,8,4,3,6,6,5,1,7,2,2,6,1,3,5,4,0,2,0,7,0
%N A387393 Decimal expansion of the imaginary part of the smallest complex solution to zeta(z) = zeta(1-z).
%C A387393 Using the reflection formula for the zeta function, one can also rewrite the equality in terms of the Gamma function as Gamma(z) = (2^(z-1))*(Pi^z)*sec((Pi*z)/2).
%C A387393 There are infinitely many solutions on the real axis and on the critical line.
%C A387393 The solutions on the critical line are the gram points and this is the first positive gram point.
%C A387393 There are 12 complex solutions apart from these out of which 3 are unique:
%C A387393    8.990914533614919... + i*4.510594140699146...
%C A387393   13.162787864991035... + i*2.580464971850669...
%C A387393   16.478090665944547... + i*0.679406009477847...
%H A387393 Wikipedia, <a href="https://en.wikipedia.org/wiki/Riemann%E2%80%93Siegel_theta_function#Gram_points">Gram points</a>
%F A387393 zeta(0.5+i*3.436218226086961...) = zeta(0.5-i*3.436218226086961...) = 0.564150979455795...
%F A387393 Smallest complex root > 0.5 of the equation Gamma(z) = (2^(z-1))*(Pi^z)*sec((Pi*z)/2).
%F A387393 Smallest positive zero of sin(theta(t)) where theta is Riemann-Siegel theta function.
%F A387393 Smallest positive root of (-0.5i)*(loggamma(.25+(i*z)*.5)-loggamma(.25-(i*z)*.5)) - (z*log(Pi))*.5 = -Pi.
%e A387393 0.5 + i*3.43621822608696159...
%t A387393 RealDigits[Im[x /. FindRoot[Zeta[x] == Zeta[1 - x], {x, 0.5+3.5I}, WorkingPrecision -> 20]]][[1]]
%Y A387393 Cf. A058303, A377302.
%K A387393 nonn,cons,new
%O A387393 1,1
%A A387393 _Jwalin Bhatt_, Aug 28 2025