ZetaZeros

In[543]:= Clear[n, s, a];
s = 1/2;
NSum[1/(n + 0)^s - 1/(n + 1)^s - 2/(n + 2)^s - 1/(n + 3)^s +
  1/(n + 4)^s + 2/(n + 5)^s, {n, 1, Infinity}, WorkingPrecision -> 400]
N[2.549127729379167407581967029267929878/(Im[ZetaZero[18]]/2), 20]^-1
N[Im[ZetaZero[1]], 20]
N[2.549127729379167407581967029267929878/(Im[ZetaZero[33]]/2), 20]^-1
N[Im[ZetaZero[2]], 20]
N[2.549127729379167407581967029267929878/(Im[ZetaZero[42]]/2), 20]^-1
N[Im[ZetaZero[3]], 20]

During evaluation of In[543]:= NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in n near {n} = {<<406>>}. NIntegrate obtained -<<406>> and <<411>> for the integral and error estimates. >>

Out[545]= \
-2.5491277293791674075819670292679298788388868361501383611646759945550\
2114281280979141892279032135882066112601786029735792734346777007416385\
1031714272022490694260100601769646158455686154700856477098022672879033\
4388858289379494842246720038280526184788343913638074859695887808089231\
7899988144451825952399138069988923352228536471316749487823829937101794\
3934028300079177278303471836107425673221464057551975

Out[546]= 14.135650568603255663
Out[547]= 14.134725141734693790
Out[548]= 21.020643640006420723
Out[549]= 21.022039638771554993
Out[550]= 25.011827067342131577
Out[551]= 25.010857580145688763


"start"
Clear[s];
s /. FindRoot[-(Zeta[s]*Zeta[s]/(2*Zeta[2*s - 1]) + (1 - 1/2^(s - 1))*
       Zeta[s] + Zeta[s]/(2*Zeta[s - 1]) + 1/Zeta[s - 1]) ==
   Im[ZetaZero[1]], {s, -1/3}, WorkingPrecision -> 50]

-0.14134053532941803058116067801572629
"end"

"start"
Clear[s];
-Limit[Zeta[s]*Zeta[s]/(2*Zeta[2*s - 1]) + (1 - 1/2^(s - 1))*Zeta[s] +
    Zeta[s]/(2*Zeta[s - 1]) + 1/Zeta[s - 1], s -> 1]
N[%, 30]
"end"

"Output:"
2 + EulerGamma - Log[4 \[Pi]]