Integrated Euler Maclaurin formula for Riemann zeta

(*Mathematica start*)
(*korrekt plot av integralen*)
(*k and q are the crucial parameters in the integrate Euler Maclaurin \
formula for the Riemann zeta function*)
Clear[q, k, r, m, i, s]
Clear[q, k, r, m, i, s, n, x, a, b, t]
k = 30;
q = 10;
Plot[Re[-I*((0*1/2 + I*t) - Sum[n^(-(1/2 + I*t))/Log[n], {n, 2, k}] +
      ExpIntegralEi[-(-1 + (1/2 + I*t)) Log[k]] +
      k^-(1/2 + I*t)/(2 Log[k]) +
      Sum[Sum[BernoulliB[2*r]/((2*r)!)*Abs[StirlingS1[2*r - 1, i]]*
         k^(-(1/2 + I*t) - 2*r + 1)*(1/2 + I*t)^i*
         Sum[-(i!/m!/(Log[k]^(i + 1 - m)*(1/2 + I*t)^(i - m))), {m, 0,
            i}], {i, 1, 2*r - 1}], {r, 1,
        q - 1}]) - (-I*((0*1/2 + I*t*0)*0 -
       Sum[n^(-(1/2 + I*t*0))/Log[n], {n, 2, k}] +
       ExpIntegralEi[-(-1 + (1/2 + I*t*0)) Log[k]] +
       k^-(1/2 + I*t*0)/(2 Log[k]) +
       Sum[Sum[BernoulliB[2*r]/((2*r)!)*Abs[StirlingS1[2*r - 1, i]]*
          k^(-(1/2 + I*t*0) - 2*r + 1)*(1/2 + I*t*0)^i*
          Sum[-(i!/m!/(Log[k]^(i + 1 - m)*(1/2 + I*t*0)^(i - m))), {m,
             0, i}], {i, 1, 2*r - 1}], {r, 1, q - 1}]))], {t, 0, 60}]



(*31.1.2017 den här fullständigt fungerande integral numeriskt \
korrekt*)
Clear[q, k, r, m, i, s, n, x, a, b, t]
k = 30;
q = 10;

s1 = N[1/2 + I*0, 20];
s = s1;
a = -I (s - Sum[n^(-s)/Log[n], {n, 2, k}] +
     ExpIntegralEi[-(-1 + s) Log[k]] + k^-s/(2 Log[k]) +
     Sum[Sum[BernoulliB[2*r]/((2*r)!)*Abs[StirlingS1[2*r - 1, i]]*
        k^(-s - 2*r + 1)*s^i*
        Sum[-(i!/m!/(Log[k]^(i + 1 - m)*s^(i - m))), {m, 0, i}], {i,
        1, 2*r - 1}], {r, 1, q - 1}]);

s2 = N[1/2 + I*60, 20];
s = s2;
b = -I (s - Sum[n^(-s)/Log[n], {n, 2, k}] +
     ExpIntegralEi[-(-1 + s) Log[k]] + k^-s/(2 Log[k]) +
     Sum[Sum[BernoulliB[2*r]/((2*r)!)*Abs[StirlingS1[2*r - 1, i]]*
        k^(-s - 2*r + 1)*s^i*
        Sum[-(i!/m!/(Log[k]^(i + 1 - m)*s^(i - m))), {m, 0, i}], {i,
        1, 2*r - 1}], {r, 1, q - 1}]);

(b - a)
NIntegrate[-I*Zeta[s], {s, s1, s2}, WorkingPrecision -> 20]
NIntegrate[Zeta[1/2 + I*t], {t, 0, 60}, WorkingPrecision -> 20]
(*end*)