关于黎曼ζ函数整数点二阶导数的公式参考及有理性问题的技术问询

我通过计算得到了黎曼ζ函数在负偶数点的二阶导数，结果如下表：

$$\begin{array}{ccc}
n & -2 n & \zeta ''(-2 n) \
1 & -2 & \frac{2 \zeta '(3)+\zeta (3) (3-2 (\gamma +\log (2 \pi )))}{4 \pi ^2} \
2 & -4 & \frac{\zeta (5) (12 (\gamma +\log (2 \pi ))-25)-12 \zeta '(5)}{8 \pi ^4} \
3 & -6 & \frac{180 \zeta '(7)-9 \zeta (7) (20 \gamma -49+20 \log (2 \pi ))}{16 \pi ^6} \
4 & -8 & \frac{9 \left(\zeta (9) (280 (\gamma +\log (2 \pi ))-761)-280 \zeta '(9)\right)}{16 \pi ^8} \
5 & -10 & \frac{45 \left(2520 \zeta '(11)+\zeta (11) (7381-2520 (\gamma +\log (2 \pi )))\right)}{32 \pi ^{10}} \
6 & -12 & \frac{135 \left(\zeta (13) (27720 (\gamma +\log (2 \pi ))-86021)-27720 \zeta '(13)\right)}{32 \pi ^{12}} \
7 & -14 & \frac{945 \left(360360 \zeta '(15)+\zeta (15) (1171733-360360 (\gamma +\log (2 \pi )))\right)}{64 \pi ^{14}} \
8 & -16 & \frac{14175 \left(\zeta (17) (720720 (\gamma +\log (2 \pi ))-2436559)-720720 \zeta '(17)\right)}{32 \pi ^{16}} \
9 & -18 & \frac{382725 \left(4084080 \zeta '(19)+\zeta (19) (14274301-4084080 (\gamma +\log (2 \pi )))\right)}{64 \pi ^{18}} \
10 & -20 & \frac{9568125 \left(\zeta (21) (15519504 (\gamma +\log (2 \pi ))-55835135)-15519504 \zeta '(21)\right)}{64 \pi ^{20}} \
\end{array}$$

基于这些计算结果，我猜想出了黎曼ζ函数在负偶数点的二阶导数公式：

$$\zeta''(-2 n)=2, \zeta'(-2 n) \left(\left(\gamma+\log (2 \pi)-H_{2 n}\right)-\frac{\zeta'(2 n+1)}{\zeta(2 n+1)}\right)$$

另外，我还计算了负奇数点的二阶导数，结果如下表：

$$\begin{array}{ccc}
n & 1-2n & \zeta''(1-2n) \
1 & -1 & \frac{\pi ^2 \left(12 \gamma (-24 \log (A)+\gamma +2)+\pi ^2+12 \left(-\log ^2(2 \pi )+\log (4)+2 \log (\pi )\right)\right)+144 \log (2 \pi ) \zeta '(2)-72 \left(\zeta ''(2)+2 \zeta '(2)\right)}{144 \pi ^2} \
2 & -3 & \frac{360 \left(3 \zeta ''(4)+(-6 \gamma +11-6 \log (2 \pi )) \zeta '(4)\right)-\pi ^6+2 \pi ^4 \left(12+6 \log ^2(2 \pi )-11 \log (4)-22 \log (\pi )+2 \gamma (3 \gamma -11+\log (64)+6 \log (\pi ))\right)}{1440 \pi ^4} \
3 & -5 & \frac{1890 \left((60 \gamma -137+60 \log (2 \pi )) \zeta '(6)-30 \zeta ''(6)\right)+5 \pi ^8-\pi ^6 \left(225+60 \log ^2(2 \pi )-137 \log (4)-274 \log (\pi )+2 \gamma (30 \gamma -137+30 \log (4)+60 \log (\pi ))\right)}{15120 \pi ^6} \
4 & -7 & \frac{170100 \left(70 \zeta ''(8)+(363-140 (\gamma +\log (2 \pi ))) \zeta '(8)\right)-105 \pi ^{10}+2 \pi ^8 \left(630 \gamma ^2-3267 \gamma +3283+630 \log ^2(2 \pi )+9 (140 \gamma -363) \log (2 \pi )\right)}{302400 \pi ^8} \
5 & -9 & \frac{374220 \left((2520 \gamma -7129+2520 \log (2 \pi )) \zeta '(10)-1260 \zeta ''(10)\right)+420 \pi ^{12}-\pi ^{10} \left(32575+5040 \log ^2(2 \pi )-14258 \log (4)-28516 \log (\pi )+4 \gamma (1260 \gamma -7129+1260 \log (4)+2520 \log (\pi ))\right)}{665280 \pi ^{10}} \
6 & -11 & \frac{12770257500 \left(13860 \zeta ''(12)+(83711-27720 (\gamma +\log (2 \pi ))) \zeta '(12)\right)-15962100 \pi ^{14}+691 \pi ^{12} \left(2096083+277200 \log ^2(2 \pi )-837110 \log (4)-1674220 \log (\pi )+20 \gamma (13860 \gamma -83711+13860 \log (4)+27720 \log (\pi ))\right)}{9081072000 \pi ^{12}} \
7 & -13 & \frac{273648375 \left((360360 (\gamma +\log (2 \pi ))-1145993) \zeta '(14)-180180 \zeta ''(14)\right)+450450 \pi ^{16}-\pi ^{14} \left(728 \left(63427+7425 \log ^2(2 \pi )\right)-17189895 \log (4)-34379790 \log (\pi )+30 \gamma (180180 \gamma -1145993+180180 \log (4)+360360 \log (\pi ))\right)}{64864800 \pi ^{14}} \
\end{array}$$

从这些结果来看，似乎有可能推导出负奇数点二阶导数的更复杂公式，但我目前还没找到头绪。不过，我在推导非负整数点的二阶导数公式时完全没有进展，所以想请教两个问题：

• 问题(1)：有没有人能提供黎曼ζ函数在整数点的二阶导数公式的参考资料？
• 问题(2)：关于$\zeta''(s)$在整数点的有理性，目前有什么已知结论吗？

备注：内容来源于stack exchange，提问作者Steven Clark