Simplify HypergeometricPFQ and MeijerG Sums?

[clrkstv (Steven Clark)]

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