A series expansion of the completed and extended Riemann zeta-function (Q2457747)

Let \[ \Xi(s):=(s-1)\zeta(s)\Gamma(\frac{s}{2}+1)\pi^{-\frac{s}{2}}. \] The author proves the formula \[ \Xi(s)=\sum_{k=0}^\infty \beta_{2k}\sum_{l=L}^\infty(-1)^l\frac{(2k+2l+2)!}{(2l)!}\frac{\zeta(2k+2l+2)}{2^{2l}}\left(\frac{1}{4l+2+s}+\frac{1}{4l+3-s}\right), \] for \(L\geq 0\), \(-2-4L<\text{Re}(s)<3+4L\), where \[ \beta_{2k}=\frac{4}{\pi}\sum_{l=0}^k\frac{(-1)^l}{(2\pi)^{2l}}\frac{(4l-1)\zeta(4l)}{(2k-2l+2)!}. \]