zeta zeros one by one maybe, with sqrt(-1) as the constant seed point

I now skipped the zeta zero asymptotic as varying seed points
and instead used s=Sqrt[-1] as the constant seed point as input to
the ratio of consecutive higher order derivatives
of the Riemann zeta function:

(*Mathematica start,takes a minute to run*)
Clear[f, s, rho, n, k, x, m];
m = 0;(*try setting:m=0,m=1,m=2,m=3,m=4,...*)
f[x_] := Zeta[x]/Sin[Pi*x/2]/
  Product[(x - ZetaZero[j])*(x - ZetaZero[-j]), {j, 1, m}];
s = I;
n = 170;
Block[{$MaxExtraPrecision = 500},
 rho = N[(1/(1 -
        Sum[((-1)^(k - 1)*Binomial[n - 1, k - 1])/
           f[k/n + s - 1/n], {k, 1, n}]/
         Sum[((-1)^(k - 1)*Binomial[n - 1, k - 1])/f[k/n + s], {k, 1,
           n}]) + s), 20]]
f[rho]
(*end*)

By changing the integer variable "m" the program
recursively factors out the zeta zeros and seems to give the
zeta zeros in order.

m=0; "gives:"
0.500000000809315424 + 14.134725141999078436 I
m=1; "gives:"
0.500003240725315219 + 21.022038108654950356 I
m=2; "gives:"
0.500044971947739555 + 25.010893102992784503 I
m=3; "gives:"
0.500605374548171145 + 30.424287649143723384 I
m=4; "gives:"
0.500830274970246344 + 32.936917518238269262 I
m=5; "gives:"
0.507131688601683165 + 37.583075616295391706 I

although to decreasing precision.

Increasing "n=170" should give more decimal digits,
but I don't have the computing power to try.



(*Mathematica start,takes a few minutes to run*)
Clear[f, s, rho, n, k, x, m];
Do[
 f[x_] :=
  Zeta[x]/Sin[Pi*x/2]/
   Product[(x - ZetaZero[j])*(x - ZetaZero[-j]), {j, 1, m}];
 n = 170;
 Print[Block[{$MaxExtraPrecision = 500},
   rho = N[(1/(1 -
          Sum[((-1)^(k - 1)*Binomial[n - 1, k - 1])/
             f[k/n + I - 1/n], {k, 1, n}]/
           Sum[((-1)^(k - 1)*Binomial[n - 1, k - 1])/f[k/n + I], {k,
             1, n}]) + I), 20]]], {m, 0, 4}]
(*end*)