Computing $\zeta (0)$

$$\begin{array}{rl}\zeta (0)& =\zeta (0,1)=\mathrm{\Gamma }(1-0)I(0,1)=1\cdot I(0,1)\\ & =\frac{1}{2\pi i}{\int }_{C_2}\frac{e^z}{z(1-e^z)}\\ & =\underset{z=0}{\mathrm{Res}}\frac{e^z}{z(1-e^z)}\\ & =\frac{1}{(2-1)!}\underset{z\to 0}{lim}\frac{d}{dz}(z-0{)}^2\frac{e^z}{z(1-e^z)}\\ & =\underset{z\to 0}{lim}\frac{d}{dz}\frac{z{e}^z}{1-e^z}\\ & =\underset{z\to 0}{lim}(z{e}^z\frac{d}{dz}\frac{1}{1-e^z}+\frac{1}{1-e^z}\frac{d}{dz}z{e}^z)\\ & =\underset{z\to 0}{lim}(z{e}^z\frac{e^z}{(1-e^z{)}^2}+\frac{1}{1-e^z}(z{e}^z+e^z))\\ & =\underset{z\to 0}{lim}\frac{z{e}^{2z}+(z{e}^z+e^z)(1-e^z)}{(1-e^z{)}^2}\\ & =\underset{z\to 0}{lim}\frac{z{e}^{2z}+z{e}^z-z{e}^{2z}+e^z-e^{2z}}{(1-e^z{)}^2}\\ & =\underset{z\to 0}{lim}\frac{e^z(1+z-e^z)}{(1-e^z{)}^2}\\ & =\underset{z\to 0}{lim}{e}^z\cdot \underset{z\to 0}{lim}\frac{1+z-e^z}{(1-e^z{)}^2}\\ & =\underset{z\to 0}{lim}\frac{1-e^z}{2(1-e^z)(-e^z)}\\ & =-\frac{1}{2}.\end{array}$$