# Zeta and gamma function

The trivial zeros of the riemman zeta function are of the form $$-2n$$ where n is a positive whole number. The gamma function goes to infinity for every negative whole number. So i want to know if there is a closed form of the limit $$\lim_{s\to -2n} \zeta(s)*\Gamma(s+1)$$ where n is a positive whole number

The Taylor series of $$\zeta(s)$$ about $$s=-2n$$ is $$\zeta(s)=\sum_{k=0}^\infty \frac{(s+2n)^k\zeta^{(k)}(-2n)}{k!}=\zeta'(-2n)(s+2n)+O((s+2n)^2)$$ we also have the Laurent series expansion of $$\Gamma(s)$$ about $$s=-2n$$ which is $$\Gamma(s)=\frac1{(2n)!(s+2n)}+O(1)$$ Multiplying these together we can find the above limit as \begin{align} \lim_{s\to-2n}\zeta(s)\Gamma(s+1) &=\lim_{s\to-2n}s\zeta(s)\Gamma(s)\\ &=\lim_{s\to-2n}s\left(\zeta'(-2n)(s+2n)+O((s+2n)^2)\right)\left(\frac1{(2n)!(s+2n)}+O(1)\right)\\ &=\lim_{s\to-2n}s\left(\frac{\zeta'(-2n)}{(2n)!}+O(s+2n)\right)\\ &=-\frac{2n\zeta'(-2n)}{(2n)!}\\ &=-\frac{\zeta'(-2n)}{(2n-1)!}\\ \end{align}