I've derived formulas for the Riemann zeta function $\zeta(s)$ and the first-order derivative $\zeta'(s)$ which are illustrated on pages 11.3 and 11.4 of the following website. I'm now attempting to understand and optimize two formulas which I've derived for the first-order derivative $\zeta'(s)$, which is the context of this question.
Illustration of Fourier Series for Prime Counting Functions

I'm having problems with evaluation times of the following two expressions where $s$ is complex, $x$ is real, $n$ is a positive integer, and the $HypergeometricPFQ$ and $MeijerG$ functions are defined in the Wolfram Language. I believe these two expressions are related as each is related to one of my two different formulas for $\zeta'(s)$. $$HypergeometricPFQ[\{1-s,1-s\},\{2-s,2-s\},-2in\pi x]+$$ $$HypergeometricPFQ[\{1-s, 1-s\},\{2-s,2-s\},2in\pi x]$$ $$MeijerG[\{\{\},\{1+s,1+s\}\},\{\{1,s,s\},\{\}\},-2in\pi x]-$$ $$MeijerG[\{\{\},\{1+s,1+s\}\},\{\{1,s,s\},\{\}\},2in\pi x]$$

Does anyone have any ideas on how I might simplify either of these two expressions to improve evaluation times?

I've found some formulas for $\zeta(s)$ that involve $HypergeometricPFQ$ and $MeijerG$ functions on the following websites, but I'm having problems finding formulas for the first-order derivative $\zeta'(s)$. Can someone point me to a website that defines formulas for the first-order derivative $\zeta'(s)$?
$\zeta(s)$ from $HypergeometricPFQ$
$\zeta(s)$ from $MeijerG$