Question (1): Is there a closed form representation for the function $f(x)$ defined in formula (1) below?
$$f(x)=\underset{N\to\infty}{\text{lim}}\left(\sum\limits_{n=1}^N \frac{(-1)^n\, (2 n+1)}{2 n!}\, \pi^n\, \frac{\zeta''(-2 n)}{\zeta'(-2 n)}\, x^{-2 n}\right)\tag{1}$$
Figure (1) below illustrates formula (1) for $f(x)$ above evaluated at $N=100$. I suspect $f(x)$ defined in formula (1) converges for $x>0$ as $N\to\infty$.
Figure (1): Illustration of formula (1) for $f(x)$
I know the values of $\zeta'(-2 n)$ are related to the values of $\zeta(2 n+1)$ as follows
$$\zeta'(-2 n)=\frac{(-1)^n\, (2 n)!\, }{2\, (2 \pi)^{2 n}}\, \zeta(2 n+1)\tag{2}$$
which leads to the equivalent formula
$$f(x)=\sum\limits_{n=1}^{\infty} \frac{4^n\, (2 n+1)}{n!\, (2 n)!}\, \pi^{3 n}\, \frac{\zeta''(-2 n)}{\zeta(2 n+1)}\, x^{-2 n}\tag{3}.$$
Question (2): Is there a relationship between $\zeta''(-2 n)$ and $\zeta'(-2 n)$ (or $\zeta(2 n+1)$) analogous to formula (2) above?
I'm not sure if it's making much progress, but I noticed the values of $\zeta''(-2n)$ seem to be related to $\zeta(2 n+1)$ and $\zeta'(2 n+1)$ as illustrated in the following table.
$$\begin{array}{cc} s & \zeta ''(s) \\ -2 & \frac{2 \zeta '(3)+\zeta (3) (3-2 (\gamma +\log (2 \pi )))}{4 \pi ^2} \\ -4 & \frac{\zeta (5) (12 (\gamma +\log (2 \pi ))-25)-12 \zeta '(5)}{8 \pi ^4} \\ -6 & \frac{180 \zeta '(7)-9 \zeta (7) (20 \gamma -49+20 \log (2 \pi ))}{16 \pi ^6} \\ -8 & \frac{9 \left(\zeta (9) (280 (\gamma +\log (2 \pi ))-761)-280 \zeta '(9)\right)}{16 \pi ^8} \\ -10 & \frac{45 \left(2520 \zeta '(11)+\zeta (11) (7381-2520 (\gamma +\log (2 \pi )))\right)}{32 \pi ^{10}} \\ -12 & \frac{135 \left(\zeta (13) (27720 (\gamma +\log (2 \pi ))-86021)-27720 \zeta '(13)\right)}{32 \pi ^{12}} \\ -14 & \frac{945 \left(360360 \zeta '(15)+\zeta (15) (1171733-360360 (\gamma +\log (2 \pi )))\right)}{64 \pi ^{14}} \\ -16 & \frac{14175 \left(\zeta (17) (720720 (\gamma +\log (2 \pi ))-2436559)-720720 \zeta '(17)\right)}{32 \pi ^{16}} \\ -18 & \frac{382725 \left(4084080 \zeta '(19)+\zeta (19) (14274301-4084080 (\gamma +\log (2 \pi )))\right)}{64 \pi ^{18}} \\ -20 & \frac{9568125 \left(\zeta (21) (15519504 (\gamma +\log (2 \pi ))-55835135)-15519504 \zeta '(21)\right)}{64 \pi ^{20}} \\ \end{array}$$
I also noticed the integer part of the denominators in the table above seems to correspond to OEIS entry A189007.
After further investigation I believe the series representation of $f(x)$ defined in formula (1) above can be split into $f(x)=g(x)+h(x)$ where
$$g(x)=\sum\limits_{n=1}^\infty \frac{(-1)^{n+1}\, (2 n+1)}{n!}\, \pi^n\, \frac{\zeta'(2 n+1)}{\zeta(2 n+1)} x^{-2 n}\tag{4}$$
which is the subject of my follow-on question and
$$h(x)=\sum\limits_{n=1}^\infty \frac{(-1)^n\, (2 n+1)}{n!}\, \pi^n \left(\gamma+\log(2 \pi)-H_{2 n}\right)\, x^{-2 n}\tag{5}$$
which I believe can be evaluated as
$$h(x)=\frac{1}{2 x^2} e^{-\frac{\pi }{x^2}} \left(x^2 \, _1F_1^{(1,0,0)}\left(0,\frac{1}{2},\frac{\pi }{x^2}\right)-2 \pi \, _1F_1^{(1,0,0)}\left(0,\frac{3}{2},\frac{\pi }{x^2}\right)+e^{\frac{\pi }{x^2}} \left(2 \pi \, _1F_1^{(1,0,0)}\left(2,2,-\frac{\pi }{x^2}\right)-x^2 \, _1F_1^{(1,0,0)}\left(1,1,-\frac{\pi }{x^2}\right)\right)-2 \left(e^{\frac{\pi }{x^2}}-1\right) x^2 (\gamma +\log (2 \pi ))+\pi (-4 \gamma +6-4 \log (2 \pi ))\right)\tag{6}.$$
This question is related to the inverse Mellin transform $\mathcal{M}_s^{-1}\left[\pi^{-\frac{s}{2}}\, (s-1)\, \Gamma\left(\frac{s}{2}+1\right) \frac{\zeta'(s)}{s\, \zeta(s)}\right]\left(\frac{1}{x}\right)$ which is clarified further in my follow-on question mentioned above.