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14,134725141734693790457251983562 = ?

N[2*Pi/ProductLog[Log[2]], 30]

Reduce[2*Pi/Log[1/n] == Im[ZetaZero[1]], n]

N[%, 30]

Reduce[2*Pi/Log[2]/n == Im[ZetaZero[1]], n]

N[%, 30]

ProductLog[Log[2]]/Log[2]

N[%, 30]

LambertW(k)/k by tetration for natural numbers.

Table[Limit[ Zeta[s]*Sum[(1 - If[Mod[k, n] == 0, n, 0])/k^(s - 1), {k, 1, n}], s -> 1], {n, 1, 12}]

Table[Limit[ Zeta[s] Total[1/Divisors[n]^(s - 1)*MoebiusMu[Divisors[n]]], s -> 1], {n, 1, 32}]

https://oeis.org/A177885

Table[Im[LogGamma[ZetaZero[n]/2]/Pi - I*Im[ZetaZero[n]]/(2*Pi)*Log[Pi] + Log[Zeta[ZetaZero[n]]]/Pi + I], {n, 1, 12}]

Plot[Im[LogGamma[1/4 + It/2]/Pi - It/(2*Pi)Log[Pi] + Log[Zeta[1/2 + It]]/Pi + I], {t, 0, 60}, ImageSize -> Large]

Round[Chop[ N[Table[Im[LogGamma[1/4 + I*t/2]]/Pi - t/(2*Pi)Log[Pi] + Im[Log[Zeta[1/2 + It]]]/Pi + 1, {t, 0, 100}]]]]

From this answer: http://math.stackexchange.com/a/442686/8530

by Raymond Manzoni.

This Excel Spreadsheet formula uses Andre LeClaire's formula to approximate the Riemann zeta zeros:

=IF(OR(ROW()=1; COLUMN()=1);0; IF(ROW()>=COLUMN();EXP(-(1-11/8/(COLUMN()-1))/EXP(1)*SUM(INDIRECT(ADDRESS(ROW()-COLUMN()+1; COLUMN(); 4)&":"&ADDRESS(ROW()-1; COLUMN(); 4); 4)));0))

(European dot-comma)

you need to divide the result with: /2/PI()/EXP(1) and take the reciprocal. tetration this is.

The von Mangoldt function matrix:

=IF(OR(ROW()=1; COLUMN()=1); 1; IF(ROW()>=COLUMN();-SUM(INDIRECT(ADDRESS(ROW()-COLUMN()+1;COLUMN(); 4)&":"&ADDRESS(ROW()-1; COLUMN(); 4); 4));-SUM(INDIRECT(ADDRESS(COLUMN()-ROW()+1;ROW(); 4)&":"&ADDRESS(COLUMN()-1; ROW(); 4); 4))))

=REPLACE(A1;FIND(".";A1);1;",")

http://pastebin.com/u/MatsGranvik

Clear[x]

x = x /. FindRoot[ 2*Pi/ProductLog[Log[x^x]] == Im[ZetaZero[1]], {x, 1.5}, WorkingPrecision -> 100]

Log[x]

(2*Pi)/Log[x]

Divisibility recurrence:

=IF(OR(COLUMN()=1); 1; IF(ROW()>=COLUMN();SUM(INDIRECT(ADDRESS(ROW()-COLUMN()+1;COLUMN()-1; 4)&":"&ADDRESS(ROW()-1; COLUMN()-1; 4); 4))-SUM(INDIRECT(ADDRESS(ROW()-COLUMN()+1;COLUMN(); 4)&":"&ADDRESS(ROW()-1; COLUMN(); 4); 4));0))

Logarithm recurrence:

=IF(OR(COLUMN()=1); 0; IF(ROW()>=COLUMN();PRODUCT(INDIRECT(ADDRESS(ROW()-COLUMN()+1;COLUMN()-1; 4)&":"&ADDRESS(ROW()-1; COLUMN()-1; 4); 4))-PRODUCT(INDIRECT(ADDRESS(ROW()-COLUMN()+1;COLUMN(); 4)&":"&ADDRESS(ROW()-1; COLUMN(); 4); 4));1))

Clear[nn];

nn = 12

f[n_, s_] = ((s + 1)^(n - 1) + s - 1)/s;

TableForm[ FullSimplify[ Table[Integrate[Integrate[f[n, s], {n, 1, 2}], {s, 0, k}], {k, 0, nn}]]]

Table[Limit[f[n, s], s -> 0], {n, 1, nn}]

Table[Limit[D[f[n, s], s], s -> 0], {n, 1, nn}]

Table[Limit[Integrate[f[-n, s], s], s -> 0], {n, 1, nn}]

FullSimplify[ Differences[Table[Limit[Sum[f[-n, s], s], s -> 0], {n, -1, nn}]]]

FullSimplify[ Differences[ Table[Limit[Sum[f[-n, s/ZetaZero[1]], s], s -> 0], {n, -1, nn}]]]