Question (1): Evaluation at non-negative even integers
Is it true that
$$\frac{\zeta '(2 n)}{\zeta (2 n)}=f(n)\tag{1}$$
where
$$f(n)=-H_{2 n-1}-\frac{\zeta'(1-2 n)}{\zeta(1-2 n)}+\gamma+\log(2 \pi)\tag{2}$$
and $n\in \mathbb{Z}\land n\geq 0$?
Question (2): Evaluation at negative odd integers
Is it also true that
$$\frac{\zeta '(-2 n-1)}{\zeta (-2 n-1)}=g(n)\tag{3}$$
where
$$g(n)=-H_{2 n+1}-\frac{\zeta '(2 n+2)}{\zeta (2 n+2)}+\gamma +\log (2 \pi )\tag{4}$$
and $n\in \mathbb{Z}\land n\geq 0$?
The last column in the following table illustrates the error in formula (1) above associated with the evaluation of $\frac{\zeta'(s)}{\zeta(s)}$ at non-negative even integers. At $n=0$ the function $f(n)$ must be evaluated as $\underset{\epsilon\to 0}{\text{lim}}f(\epsilon)$.
$$\begin{array}{ccccc} n & 2 n & \frac{\zeta'(2 n)}{\zeta(2 n)} & f(n) & f(n)-\frac{\zeta'(2 n)}{\zeta(2 n)} \\ 0 & 0 & \log (2 \pi ) & \log (2 \pi ) & 0. \\ 1 & 2 & \frac{6 \zeta '(2)}{\pi ^2} & 12 \left(\frac{1}{12}-\log (A)\right)-1+\gamma+\log (2 \pi ) & 0. \\ 2 & 4 & \frac{90 \zeta '(4)}{\pi ^4} & -120 \zeta '(-3)-\frac{11}{6}+\gamma +\log (2 \pi ) & \text{5.551115123125783$\grave{ }$*${}^{\wedge}$-16} \\ 3 & 6 & \frac{945 \zeta '(6)}{\pi ^6} & 252 \zeta '(-5)-\frac{137}{60}+\gamma +\log (2 \pi ) & \text{2.220446049250313$\grave{ }$*${}^{\wedge}$-16} \\ 4 & 8 & \frac{9450 \zeta '(8)}{\pi ^8} & -240 \zeta '(-7)-\frac{363}{140}+\gamma +\log (2 \pi ) & \text{3.0531133177191805$\grave{ }$*${}^{\wedge}$-16} \\ 5 & 10 & \frac{93555 \zeta '(10)}{\pi ^{10}} & 132 \zeta '(-9)-\frac{7129}{2520}+\gamma +\log (2 \pi ) & \text{3.3306690738754696$\grave{ }$*${}^{\wedge}$-16} \\ 6 & 12 & \frac{638512875 \zeta '(12)}{691 \pi ^{12}} & -\frac{32760}{691} \zeta '(-11)-\frac{83711}{27720}+\gamma +\log (2 \pi ) & \text{6.661338147750939$\grave{ }$*${}^{\wedge}$-16} \\ 7 & 14 & \frac{18243225 \zeta '(14)}{2 \pi ^{14}} & 12 \zeta '(-13)-\frac{1145993}{360360}+\gamma +\log (2 \pi ) & \text{1.2212453270876722$\grave{ }$*${}^{\wedge}$-15} \\ 8 & 16 & \frac{325641566250 \zeta '(16)}{3617 \pi ^{16}} & -\frac{8160 \zeta '(-15)}{3617}-\frac{1195757}{360360}+\gamma +\log (2 \pi ) & \text{1.9984014443252818$\grave{ }$*${}^{\wedge}$-15} \\ 9 & 18 & \frac{38979295480125 \zeta '(18)}{43867 \pi ^{18}} & \frac{14364 \zeta '(-17)}{43867}-\frac{42142223}{12252240}+\gamma +\log (2 \pi ) & \text{3.1086244689504383$\grave{ }$*${}^{\wedge}$-15} \\ 10 & 20 & \frac{1531329465290625 \zeta '(20)}{174611 \pi ^{20}} & -\frac{6600 \zeta '(-19)}{174611}-\frac{275295799}{77597520}+\gamma +\log (2 \pi ) & \text{2.6645352591003757$\grave{ }$*${}^{\wedge}$-15} \\ \end{array}$$
The last column in the following table illustrates the error in formula (3) above associated with the evaluation of $\frac{\zeta'(s)}{\zeta(s)}$ at negative odd integers.
$$\begin{array}{ccccc} n & -2n-1 & \frac{\zeta'(-2n-1)}{\zeta(-2n-1)} & g(n) & g(n)-\frac{\zeta'(-2n-1)}{\zeta(-2n-1)} \\ 0 & -1 & -12 \left(\frac{1}{12}-\log (A)\right) & -\frac{6 \zeta '(2)}{\pi ^2}-1+\gamma+\log (2 \pi ) & 0. \\ 1 & -3 & 120 \zeta '(-3) & -\frac{90 \zeta '(4)}{\pi ^4}-\frac{11}{6}+\gamma +\log (2 \pi ) & \text{5.551115123125783$\grave{ }$*${}^{\wedge}$-16} \\ 2 & -5 & -252 \zeta '(-5) & -\frac{945 \zeta '(6)}{\pi ^6}-\frac{137}{60}+\gamma +\log (2 \pi ) & \text{2.220446049250313$\grave{ }$*${}^{\wedge}$-16} \\ 3 & -7 & 240 \zeta '(-7) & -\frac{9450 \zeta '(8)}{\pi ^8}-\frac{363}{140}+\gamma +\log (2 \pi ) & \text{3.0531133177191805$\grave{ }$*${}^{\wedge}$-16} \\ 4 & -9 & -132 \zeta '(-9) & -\frac{93555 \zeta '(10)}{\pi ^{10}}-\frac{7129}{2520}+\gamma +\log (2 \pi ) & \text{3.3306690738754696$\grave{ }$*${}^{\wedge}$-16} \\ 5 & -11 & \frac{32760}{691} \zeta '(-11) & -\frac{638512875 \zeta '(12)}{691 \pi ^{12}}-\frac{83711}{27720}+\gamma +\log (2 \pi ) & \text{6.661338147750939$\grave{ }$*${}^{\wedge}$-16} \\ 6 & -13 & -12 \zeta '(-13) & -\frac{18243225 \zeta '(14)}{2 \pi ^{14}}-\frac{1145993}{360360}+\gamma +\log (2 \pi ) & \text{1.2212453270876722$\grave{ }$*${}^{\wedge}$-15} \\ 7 & -15 & \frac{8160 \zeta '(-15)}{3617} & -\frac{325641566250 \zeta '(16)}{3617 \pi ^{16}}-\frac{1195757}{360360}+\gamma +\log (2 \pi ) & \text{1.9984014443252818$\grave{ }$*${}^{\wedge}$-15} \\ 8 & -17 & -\frac{14364 \zeta '(-17)}{43867} & -\frac{38979295480125 \zeta '(18)}{43867 \pi ^{18}}-\frac{42142223}{12252240}+\gamma +\log (2 \pi ) & \text{3.1086244689504383$\grave{ }$*${}^{\wedge}$-15} \\ 9 & -19 & \frac{6600 \zeta '(-19)}{174611} & -\frac{1531329465290625 \zeta '(20)}{174611 \pi ^{20}}-\frac{275295799}{77597520}+\gamma +\log (2 \pi ) & \text{2.6645352591003757$\grave{ }$*${}^{\wedge}$-15} \\ 10 & -21 & -\frac{276 \zeta '(-21)}{77683} & -\frac{13447856940643125 \zeta '(22)}{155366 \pi ^{22}}-\frac{18858053}{5173168}+\gamma +\log (2 \pi ) & \text{8.881784197001252$\grave{ }$*${}^{\wedge}$-15} \\ \end{array}$$
It is known that
$$ \zeta(1-s)=2^{1-s}\pi^{-s}\cos\left(\pi s\over2\right)\Gamma(s)\zeta(s) $$
Taking logarithmic derivative on both side, we have
$$ -{\zeta'\over\zeta}(1-s)=-\log2\pi-\frac\pi2\tan\left(\pi s\over2\right)+{\Gamma'\over\Gamma}(s)+{\zeta'\over\zeta}(s) $$
When $s=2n\ge0$, we have $\tan(n\pi)=0$, so
$$ -{\zeta'\over\zeta}(1-2n)=-\log2\pi+{\Gamma'\over\Gamma}(2n)+{\zeta'\over\zeta}(2n) $$
For the Gamma term, plug in Weierstrass product that
$$ \Gamma(s)={e^{-s\gamma}\over s}\prod_{k=1}^\infty\left(1+\frac sk\right)^{-1}e^{s/k} $$
we have
$$ {\Gamma'\over\Gamma}(s)=-\gamma-\frac1s+\sum_{k=1}^\infty\left(\frac1k-{1\over s+k}\right) $$
so that
$$ \begin{aligned} {\Gamma'\over\Gamma}(2n) &=-\gamma-{1\over2n}+\sum_{k=1}^\infty\left(\frac1k-{1\over2n+k}\right) \\ &=-\gamma-{1\over2n}+\sum_{k=1}^{2n}\frac1k=-\gamma+H_{2n-1} \end{aligned} $$
Plugging this into the original equation, we get
$$ -{\zeta'\over\zeta}(1-2n)=-\log2\pi-\gamma+H_{2n-1}+{\zeta'\over\zeta}(2n) $$
Rearranging this equation gives
$$ {\zeta'\over\zeta}(2n)=-H_{2n-1}-{\zeta'\over\zeta}(1-2n)+\gamma+\log2\pi $$
Similar procedure can be done for $n<0$ by using
$$ \zeta(s)=2^s\pi^{s-1}\sin\left(\pi s\over2\right)\Gamma(1-s)\zeta(1-s) $$