Most zeta zeros by two step iterative bisection.

Clear[x, k, c, numberOfIterations, i, t, x1, x2, f, a, b]
f[a_] = 2*Pi*E^1*
   E^LambertW[((a/(2*Pi))*Log[a/(2*Pi*E)] +
        Arg[Zeta[j]/Zeta[1/2 + I*a + j - 1]]/Pi - c + k -
        RiemannSiegelTheta[a]/Pi)/E^1];
numberOfIterations = 20;
(*Interesting values of c are:c=0,c=1/2,c=1/4,c=3/4*)
(*c=0 gives Gram points*)
(*c=1/2 gives Franca-LeClair points*)
(*c=1/4 gives non-zero self \
intersections:Re[Zeta[1/2+I*t]]=Im[Zeta[1/2+I*t]]*)
(*c=3/4 gives:Re[Zeta[1/2+I*t]]=-Im[Zeta[1/2+I*t]]*)
c = 1/2;
j = 1 + 1/10^40;
nn = 137;
Monitor[Table[
  a = 2*Pi*Exp[1]*Exp[ProductLog[(k + 1 - 11/8)/Exp[1]]];
  b = a;
  Table[
   a = N[Round[f[b], 10^-15], 14];
   b = N[Round[f[a], 10^-15], 14];
   {a, b}, {i, 1, numberOfIterations + 11}];
  z = Table[
    cc = (N[Round[a, 10^-15], 14] + N[Round[b, 10^-15], 14])/2;
    s = Sign[b - a]*Sign[f[cc] - cc];
    a = (1 + s)/2*cc + (1 - s)/2*a;
    b = (1 - s)/2*cc + (1 + s)/2*b;
    {a, b}, {i, 1, numberOfIterations + 11}];
  Mean[z[[numberOfIterations + 10]]](*,
  N[Im[ZetaZero[k+1]],14]*), {k, 0, nn - 1}], k]
% - Im[ZetaZero[Range[nn]]]