Editing Riemann zeta function (section)

===Hadamard product===
On the basis of [[Weierstrass factorization theorem|Weierstrass's factorization theorem]], [[Hadamard]] gave the [[infinite product]] expansion

:<math>\zeta(s) = \frac{e^{\left(\log(2\pi)-1-\frac{\gamma}{2}\right)s}}{2(s-1)\Gamma\left(1+\frac{s}{2}\right)} \prod_\rho \left(1 - \frac{s}{\rho} \right) e^\frac{s}{\rho},</math>

where the product is over the non-trivial zeros {{mvar|ρ}} of {{math|''ζ''}} and the letter {{mvar|γ}} again denotes the [[Euler–Mascheroni constant]]. A simpler [[infinite product]] expansion is

:<math>\zeta(s) = \pi^\frac{s}{2} \frac{\prod_\rho \left(1 - \frac{s}{\rho} \right)}{2(s-1)\Gamma\left(1+\frac{s}{2}\right)}.</math>

This form clearly displays the simple pole at {{math|''s'' {{=}} 1}}, the trivial zeros at −2,&nbsp;−4,&nbsp;... due to the gamma function term in the denominator, and the non-trivial zeros at {{math|''s'' {{=}} ''ρ''}}. (To ensure convergence in the latter formula, the product should be taken over "matching pairs" of zeros, i.e. the factors for a pair of zeros of the form {{mvar|ρ}} and {{math|1 − ''ρ''}} should be combined.)