It is known that $$\Gamma (s) \zeta (s)=\int_0^{\infty} \frac{x^{s-1}}{e^x-1}dx$$ this function is valid only for $\Re{s}>1$.

However, if we ignore this restriction, and integrate by using $$\frac{x}{e^x-1}=\sum_{n=0}^{\infty}\frac{B_n}{n!}x^n$$ we could get $$\Gamma (s) \zeta (s)=\frac{1}{s-1}-\frac{1}{2s}+\sum_{n=2}^{\infty}\frac{B_n}{n!(s+n-1)}+\int_1^{\infty} \frac{x^{s-1}}{e^x-1}dx$$ Surprisly, this expression coincides with the true value of $\zeta(s)$ precisly in the compelx region $$0\leq\Re(s)\leq 1,-10\leq\Im(s)\leq10$$ Any explanations?