I have proved the following four identities already but I can't prove $\zeta(0)=-\frac12$. The book has stated it as "an easy corollary". I couldn't find any complete proof in the internet because either there is none or uses formulas not from the following four (and not based on this):
How $\zeta(0)=-\frac12$ is true?
Take $s \rightarrow 0$. The LHS is equal to $\frac{2}{s}(\zeta(0)+o(1))$ while the RHS is equivalent to $-\pi^{-1/2}\Gamma(1/2)\frac{1}{s}$. But $\Gamma(1/2)=\pi^{1/2}$, thus $2\zeta(0)+o(1) = -1$, hence the conclusion.