Are there iterative formulas to find zeta zeros? I am wondering whether one could find Riemann zeta zeros iteratively by using relationships such as this one:
$$\rho _1=\lim_{s\to 1} \, \frac{\zeta (s) \zeta \left(s \cdot \rho _1\right)}{\zeta '(\rho _1)}$$
where $\rho _1 \approx 0.5 + 14.1347i$ is the first zeta zero.
Are there better relationships like Newton iteration or something like that?
The relation above that I tried to use diverges when trying to iterate it.
 A: Yes, if the Riemann hypothesis is true, see https://github.com/crowlogic/fastmath/raw/master/research/ZCauchy.pdf where the link is for the 4th version of the paper at http://vixra.org/abs/1702.0273  . If the there is zero of Hardy Z with nonzero imaginary part then the iteration will not reach it when started from the real line, because the complex conjugate pair would cancel their mutual influences on the trajectories of the Y iteration. If its possible to prove Im(Y_n_m) always has decreasing imaginary magnitude when started with non-zero imaginary part then that would prove that it would never converge to a zero with nonzero imaginary part because it didn't exist.
More concisely, 
Let $Y^- (t) = t - \tanh (Z (t))$ and $Y^+ (t) = t + \tanh (Z (t))$ where $Z(t)$ is the Hardy Z function, then, if the Riemann hypothesis is true then $Y^- (t)$ has attractive fixed-points at the odd-numbered zeros $Z(t_n)=0$ and repelling fixed-points at the even-numbered zeros $Z(t_n)=0$ and vice versa for $Y^+ (t)$  which would have attractive fixed-points at the even-numbered zeros and repulsive fixed-points at the odd-numbered zeros. 
The multiplier $\lambda_f (\alpha)$ of a fixed point $\alpha$ of a map $f (x)$ is equal to the derivative $\dot{f} (\alpha)$ of the map evaluated at the point $\alpha$ which is the first term in the Taylor expansion at that point
\begin{equation}
  \lambda_f (\alpha) = \dot{f} (\alpha)
\end{equation}
If $| \lambda_f (\alpha) | < 1$ then $\alpha$ is a said to be an attractive
fixed-point of $f (x)$. If $| \lambda_f (\alpha) | = 1$ then $\alpha$ is an
indifferent fixed-point of $f (t)$ also known as as neutral fixed-point, and if $| \lambda_f (\alpha) > 1 |$ then $\alpha$ is a repelling fixed-pint of $f(t)$. When $| \lambda_f (\alpha) | = 0$ the fixed-point $\alpha$ is said to be superattractive fixed-point of $f (t)$.
this method only converges to simple zeros. multiple zeros of any funxtion f(t) correspond to indifferent fixed points of t-tanh(f(t)) and t+tanh(f(t)). 
there is also an indifferent fixed point of Yplus and Yminus corresponding to the pole of Z at -i/2
 as well as indifferent fixed points at the trivial roots. poles of Z cause tanh to reach its limiting value at plus or minus one and the derivative of t plus or minus one is 1
The tight span, aka hyperconvex hull, or injective envelope of Y is H(t)=[t-1,t+1]

The homotopy of composed mappings is visualized here


