%I #16 Nov 23 2018 09:22:06

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

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

%U 8,1,5,6,0,8,6,3,0,5,5,1,0,0,5,9,3,8,8,4,8,4,9,6,1,3,5,3

%N Decimal expansion of imaginary part of 5th nontrivial zero of Riemann zeta function.

%C The real part of the 5th nontrivial zero is of course 1/2 (A020761; the Riemann hypothesis is here assumed to be true).

%H Andrew M. Odlyzko, <a href="http://www.plouffe.fr/simon/constants/zeta100.html">The first 100 (non trivial) zeros of the Riemann Zeta function, to over 1000 decimal digits each</a>, AT&T Labs - Research.

%H Andrew M. Odlyzko, <a href="http://www.dtc.umn.edu/~odlyzko/zeta_tables/index.html">Tables of zeros of the Riemann zeta function</a>

%e The zero is at 1/2 + i * 32.9350615877391896906623689640749...

%t (* ZetaZero was introduced in Version 6.0 *) RealDigits[ZetaZero[5], 10, 100][[1]]

%o (PARI) solve(y=32,33,real(zeta(1/2+y*I))) \\ _Charles R Greathouse IV_, Mar 10 2016

%o (PARI) lfunzeros(lzeta,[32,33])[1] \\ _Charles R Greathouse IV_, Mar 10 2016

%Y Cf. A002410: nearest integer to imaginary part of n-th zero of Riemann zeta function (main entry); also A013629 (floor) and A092783 (ceiling).

%Y The imaginary parts of the first 4 zeros are 14.134725... (A058303), 21.0220396... (A065434), 25.01085758... (A065452), 30.424876... (A065453). Others are A305741 (k=6), A305742 (k=7), A305743 (k=8), A305744 (k=9), A306004 (k=10).

%Y The real parts of the trivial zeros are given by A005843 multiplied by -1 (and ignoring the initial 0 of that sequence).

%K nonn,cons

%O 2,1

%A _Alonso del Arte_, Jul 02 2011

%E Example and cross-references edited by _M. F. Hasler_, Nov 23 2018