The following table of evaluations
$$\begin{array}{ccc} n & -2 n & \zeta ''(-2 n) \\ 1 & -2 & \frac{2 \zeta '(3)+\zeta (3) (3-2 (\gamma +\log (2 \pi )))}{4 \pi ^2} \\ 2 & -4 & \frac{\zeta (5) (12 (\gamma +\log (2 \pi ))-25)-12 \zeta '(5)}{8 \pi ^4} \\ 3 & -6 & \frac{180 \zeta '(7)-9 \zeta (7) (20 \gamma -49+20 \log (2 \pi ))}{16 \pi ^6} \\ 4 & -8 & \frac{9 \left(\zeta (9) (280 (\gamma +\log (2 \pi ))-761)-280 \zeta '(9)\right)}{16 \pi ^8} \\ 5 & -10 & \frac{45 \left(2520 \zeta '(11)+\zeta (11) (7381-2520 (\gamma +\log (2 \pi )))\right)}{32 \pi ^{10}} \\ 6 & -12 & \frac{135 \left(\zeta (13) (27720 (\gamma +\log (2 \pi ))-86021)-27720 \zeta '(13)\right)}{32 \pi ^{12}} \\ 7 & -14 & \frac{945 \left(360360 \zeta '(15)+\zeta (15) (1171733-360360 (\gamma +\log (2 \pi )))\right)}{64 \pi ^{14}} \\ 8 & -16 & \frac{14175 \left(\zeta (17) (720720 (\gamma +\log (2 \pi ))-2436559)-720720 \zeta '(17)\right)}{32 \pi ^{16}} \\ 9 & -18 & \frac{382725 \left(4084080 \zeta '(19)+\zeta (19) (14274301-4084080 (\gamma +\log (2 \pi )))\right)}{64 \pi ^{18}} \\ 10 & -20 & \frac{9568125 \left(\zeta (21) (15519504 (\gamma +\log (2 \pi ))-55835135)-15519504 \zeta '(21)\right)}{64 \pi ^{20}} \\ \end{array}$$
led to the conjectured formula
$$\zeta''(-2 n)=2\, \zeta'(-2 n) \left(\left(\gamma+\log (2 \pi)-H_{2 n}\right)-\frac{\zeta'(2 n+1)}{\zeta(2 n+1)}\right)$$
for the second-order derivative of the Riemann zeta function at negative even integers and the following table of evaluations
$$\begin{array}{ccc} n & 1-2n & \zeta''(1-2n) \\ 1 & -1 & \frac{\pi ^2 \left(12 \gamma (-24 \log (A)+\gamma +2)+\pi ^2+12 \left(-\log ^2(2 \pi )+\log (4)+2 \log (\pi )\right)\right)+144 \log (2 \pi ) \zeta '(2)-72 \left(\zeta ''(2)+2 \zeta '(2)\right)}{144 \pi ^2} \\ 2 & -3 & \frac{360 \left(3 \zeta ''(4)+(-6 \gamma +11-6 \log (2 \pi )) \zeta '(4)\right)-\pi ^6+2 \pi ^4 \left(12+6 \log ^2(2 \pi )-11 \log (4)-22 \log (\pi )+2 \gamma (3 \gamma -11+\log (64)+6 \log (\pi ))\right)}{1440 \pi ^4} \\ 3 & -5 & \frac{1890 \left((60 \gamma -137+60 \log (2 \pi )) \zeta '(6)-30 \zeta ''(6)\right)+5 \pi ^8-\pi ^6 \left(225+60 \log ^2(2 \pi )-137 \log (4)-274 \log (\pi )+2 \gamma (30 \gamma -137+30 \log (4)+60 \log (\pi ))\right)}{15120 \pi ^6} \\ 4 & -7 & \frac{170100 \left(70 \zeta ''(8)+(363-140 (\gamma +\log (2 \pi ))) \zeta '(8)\right)-105 \pi ^{10}+2 \pi ^8 \left(630 \gamma ^2-3267 \gamma +3283+630 \log ^2(2 \pi )+9 (140 \gamma -363) \log (2 \pi )\right)}{302400 \pi ^8} \\ 5 & -9 & \frac{374220 \left((2520 \gamma -7129+2520 \log (2 \pi )) \zeta '(10)-1260 \zeta ''(10)\right)+420 \pi ^{12}-\pi ^{10} \left(32575+5040 \log ^2(2 \pi )-14258 \log (4)-28516 \log (\pi )+4 \gamma (1260 \gamma -7129+1260 \log (4)+2520 \log (\pi ))\right)}{665280 \pi ^{10}} \\ 6 & -11 & \frac{12770257500 \left(13860 \zeta ''(12)+(83711-27720 (\gamma +\log (2 \pi ))) \zeta '(12)\right)-15962100 \pi ^{14}+691 \pi ^{12} \left(2096083+277200 \log ^2(2 \pi )-837110 \log (4)-1674220 \log (\pi )+20 \gamma (13860 \gamma -83711+13860 \log (4)+27720 \log (\pi ))\right)}{9081072000 \pi ^{12}} \\ 7 & -13 & \frac{273648375 \left((360360 (\gamma +\log (2 \pi ))-1145993) \zeta '(14)-180180 \zeta ''(14)\right)+450450 \pi ^{16}-\pi ^{14} \left(728 \left(63427+7425 \log ^2(2 \pi )\right)-17189895 \log (4)-34379790 \log (\pi )+30 \gamma (180180 \gamma -1145993+180180 \log (4)+360360 \log (\pi ))\right)}{64864800 \pi ^{14}} \\ \end{array}$$
seems to imply it might be possible to derive a more complicated formula for the second-order derivative of the Riemann zeta function at negative odd integers.
But I can't seem to make any progress on formulas for the second-order derivative of the Riemann zeta function at non-negative integers.
Question (1): Can anyone provide a reference for formulas for the second-order derivative of the Riemann zeta function at integer values?
Question (2): Is anything known about the rationality of $\zeta''(s)$ at integer values of $s$?
