Normalization of the Riemann zeta function.

Clear[x, n, c, f, g]
c = 1/2;
f[x_] = 2*Pi*E*
   E^LambertW[((N[x]/(2*Pi))*Log[N[x]/(2*Pi*E)] - c + n -
        RiemannSiegelTheta[N[x]]/Pi)/E];
g[y_] = f[f[f[f[y]]]];
Show[Monitor[
  Table[ListLinePlot[
    Table[Re[Zeta[1/2 + I*g[1]]], {n, k, k + 1, 1/80}],
    DataRange -> {0, 1}, PlotRange -> {-1.3, 6.4}], {k, 0, 100}], k]]
Clear[x, n, c, f, g]
c = 1/2;
f[x_] = 2*Pi*E^1*
   E^LambertW[((x/(2*Pi))*Log[x/(2*Pi*E)] - c + n -
        RiemannSiegelTheta[x]/Pi)/E^1];
g[y_] = f[f[f[f[y]]]];
Show[Monitor[
  ListLinePlot[
   Sum[Table[Re[Zeta[1/2 + I*g[1]]], {n, k, k + 1, 1/80}], {k, 0,
      100}]/101, DataRange -> {0, 1}], k]]