Newton Raphson iteration applied to Riemann zeta function

(*Newton Raphson iteration*)
(*start*)
f[t_] = Zeta[1/2 + I*t]/Zeta'[1/2 + I*t];
g[t_] = Im[(1/2 + I*t) - f[t]];
min = 0;
max = 60;
Show[Plot[g[g[g[g[g[g[g[g[g[g[g[t]]]]]]]]]]], {t, min, max}],
 Plot[t, {t, min, max}],
 Table[Graphics[{PointSize[Large],
    Point[{Im[ZetaZero[n]], Im[ZetaZero[n]]}]}], {n, 1, 14}]]
(*end*)

(* variant of logarithmic derivative 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 = 0;
max = 60;
Show[Plot[g[g[g[g[g[g[t]]]]]], {t, min, max}], Plot[t, {t, min, max}],
  Table[Graphics[{PointSize[Large],
    Point[{Im[ZetaZero[n]], Im[ZetaZero[n]]}]}], {n, 1, 14}]]
(*end*)