So I’ve read Weierstrass’ factorization theorem. I sort of know how the Hadamard product of the zeta function was derived. I don’t get however, how they moved from this

$$\zeta(s)=\frac{e^{(\ln(2\pi) -1-\gamma /2)s}}{2(s-1)\Gamma (1+\frac{s}{2})} \prod_{\rho} (1-\frac{s}{\rho})e^{s/ \rho}$$

To this simpler and more elegant product

$$\zeta(s)=\frac{\pi ^{s/2}}{2(s-1) \Gamma (1+\frac{s}{2})} \prod_{\rho} (1-\frac{s}{\rho})$$

Where $\rho$ are the roots of the zeta function.

In the case of $sin(x)$ I get how

$$sin(x)=x\prod_{n=1}^{\infty} (1-\frac{s}{\pi n})e^{s/ \pi n} = x\prod_{n=1}^{\infty} (1-(\frac{s}{\pi n})^2)$$

Since the function has zeros located symmetrically about the origin and therefore can be paired with the opposite zero which cancels the exponential in the product.

However the zeta function does not have that property. It is not evident how the right side of the equation follows from the left. Hence my question, how do I show that the right hand side follows from the Hadamard product?


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