Added Background: The pair correlation of the zeros of the Riemann zeta function is influenced by the the derivative of the logarithmic derivative $(\zeta^\prime(s)/\zeta(s))^\prime$; see for example the answers to this question

I'm looking for references for computation of $\zeta^\prime(s)/\zeta(s)$, as well as the its derivative, in the critical strip $0<\text{Re}(s)<1$. (I'm sure they're out there but google scholar/MathSciNet searches return way too many irrelevant hits.)

Of course, both $\zeta(s)$ and $\zeta^\prime(s)$ are implemented in packages like Sage, one can just take the quotient and then use this to numerically estimate the derivative via the difference quotient, but this seems computationally wasteful. We have that $$ \frac{\zeta^\prime(s)}{\zeta(s)}=\log(2\pi)-1-\gamma/2-\frac{1}{s-1}-\frac12\frac{\Gamma^\prime(s/2+1)}{\Gamma(s/2+1)}+\sum_\rho\left(\frac{1}{s-\rho}+\frac{1}{\rho}\right). $$

Similarly, one gets $(\zeta^\prime(s)/\zeta(s))^\prime$ upon differentiating term by term.

The digamma function $\Gamma^\prime/\Gamma$ as well as the Riemann zeros $\rho$ are implemented in *Mathematica*. So I think what I'm asking is a reference to answer the following:

Given $\epsilon$, how many zeros do I need to take as a as a function of $t=\text{Im}(s)$ so the error is bounded by $\epsilon$?

Added: In Theorem 9.6(A) in Titchmarsh's "Theory of the Riemann Zeta Function", one can compute the relevant constants to show that $$ \left|\frac{\zeta^\prime(s)}{\zeta(s)}-\sum_{|\rho-s|\le 6}\frac{1}{s-\rho}\right|\le 4\log t. $$ So in answer to Joro's question below, yes the sum is dominated by the zeros close to $s$.

A naive implementation, using all the zeros up to height $2t$, gives the following graphic for the $\arg(\zeta^\prime/\zeta)$, interpreted as a color. Here $0<\sigma<1$, $1000< t<1010$.