Attempt at finding zeta zeros by recurrence of zeta function.

Question

The following Mathematica program converges to most of the riemann zeta zeros, by using an approximation as a starting point.

Clear[n, k, a]
Do[a = 1/2 + N[I*2*Pi*(n - 11/8)/LambertW[(n - 11/8)/Exp[1]], 100];
 Do[a = a - Im[Zeta[a]]*I;,
   {k, 1, 80}] 
  Print[a],
 {n, 1, 12}]

Where the key statement is the iteration of:

a = a - Im[Zeta[a]]*I
a = a - Im[Zeta[a]]*I
a = a - Im[Zeta[a]]*I
a = a - Im[Zeta[a]]*I

80 times.

The approximation is the one by Andre LeClair in a paper on arXiv.

Out comes the values of the zeta zeros, but it misses a few and doubles some due to the starting values in the LambertW approximation.

$$1/2+14.134725141734693790457251983562 I$$ $$1/2+21.022039638771554992628479593896902777334 I$$ $$1/2+25.0108575801456887632137909925628218186595499 I$$ $$1/2+30.42487612585951321031189753058409132018156002372 I$$

Is this accurate enough to approach to the Riemann hypothesis?