Limiting ratios giving zeta zeros vs Newton Raphson

(*start apparently equivalent to Newton Raphson*)
cons = 9;
ww = 400;
div = 10;
Monitor[TableForm[zz = Table[Clear[t, b, n, k, nn, x];
     z = N[cons + w/div, 20];
     polynomial = Normal[Series[Zeta[(x + z*N[I, 20])], {x, 0, 10}]];
     digits = 20;
     b = With[{nn = 20},
       CoefficientList[Series[1/polynomial, {x, 0, nn}], x]];
     nn = Length[b] - 1;
     x = z*I + N[b[[nn - 1]]/b[[nn]], digits], {w, 0, ww}]];, w]
g1 = ListLinePlot[Flatten[Im[zz]], DataRange -> {cons, cons + ww/div}]
g2 = ListLinePlot[Flatten[Re[zz]], DataRange -> {cons, cons + ww/div},
   PlotRange -> {-2, 2}]
zz
(*end apparently equivalent to Newton Raphson*)

(* approximation of logarithmic derivative giving Newton Raphson \
iteration*)
(*start*)
c = 1 + 1/1000;
f[t_] = -1/((Zeta[1/2 + I*t]*Zeta[c])/Zeta[1/2 + I*t + c - 1] -
     Zeta[c]);
g[t_] = Im[(1/2 + I*t) - f[t]];
min = cons;
max = cons + ww/div;
g2 = Show[Plot[g[g[g[g[g[g[t]]]]]], {t, min, max}, PlotStyle -> Red],
  Table[Graphics[{PointSize[Large],
     Point[{Im[ZetaZero[n]], Im[ZetaZero[n]]}]}], {n, 1, 14}]]

Show[g1, g2]
(*end*)