(1)

\begin{align*} \eta\left(s\right) & =-\sum_{k=1}^{\infty}\frac{\left(-1\right)^{k}}{k^{s}}\\ & =-\frac{1}{2}-\frac{1}{2}\sum_{k=1}^{\infty}\left(\frac{\left(-1\right)^{k}}{k^{s}}+\frac{\left(-1\right)^{k+1}}{\left(k+1\right)^{s}}\right) \end{align*}

\[ \eta'\left(s\right)=\frac{1}{2}\sum_{k=1}^{\infty}\left(\frac{\left(-1\right)^{k}}{k^{s}}\Log k+\frac{\left(-1\right)^{k+1}}{\left(k+1\right)^{s}}\Log\left(k+1\right)\right) \]

より、

\begin{align*} \eta'\left(0\right) & =\frac{1}{2}\sum_{k=1}^{\infty}\left(\left(-1\right)^{k}\Log k+\left(-1\right)^{k+1}\Log\left(k+1\right)\right)\\ & =-\frac{1}{2}\sum_{k=1}^{\infty}\left(\left(-1\right)^{k}\Log\frac{k+1}{k}\right)\\ & =-\frac{1}{2}\sum_{k=1}^{\infty}\left(\left(-1\right)^{2k-1}\Log\frac{\left(2k-1\right)+1}{\left(2k-1\right)}+\left(-1\right)^{2k}\Log\frac{2k+1}{2k}\right)\\ & =\frac{1}{2}\sum_{k=1}^{\infty}\left(\Log\frac{\left(2k\right)\left(2k\right)}{\left(2k-1\right)\left(2k+1\right)}\right)\\ & =\frac{1}{2}\Log\prod_{k=1}^{\infty}\frac{\left(2k\right)\left(2k\right)}{\left(2k-1\right)\left(2k+1\right)}\\ & =\frac{1}{2}\Log\frac{\pi}{2}\cmt{\text{ウォリス積}}\\ & =\Log\sqrt{\frac{\pi}{2}} \end{align*}

(2)

\begin{align*} \zeta'\left(0\right) & =\left[\frac{d}{ds}\zeta\left(s\right)\right]_{s=0}\\ & =\left[\frac{d}{ds}\frac{\eta\left(s\right)}{1-2^{1-s}}\right]_{s=0}\\ & =\left[\frac{\eta'\left(s\right)\left(1-2^{1-s}\right)-\eta\left(s\right)\left(2^{1-s}\Log2\right)}{\left(1-2^{1-s}\right)^{2}}\right]_{s=0}\\ & =-\left(\eta'\left(0\right)+2\eta\left(0\right)\Log2\right)\\ & =-\left(\Log\sqrt{\frac{\pi}{2}}+2\left(\frac{1}{2}\right)\Log2\right)\\ & =-\left(\Log\sqrt{\frac{\pi}{2}}+\Log2\right)\\ & =-\Log\sqrt{2\pi} \end{align*}