The Hadamard product in complex analysis is a representation for the Riemann zeta function 
as a product over its nontrivial
zeros 
,
![zeta(s)=(e^([ln(2pi)-1-gamma/2]s))/(2(s-1)Gamma(1+1/2s))product_(rho)(1-s/rho)e^(s/rho),](/images/equations/HadamardProduct/NumberedEquation1.svg) |
(1)
|
where 
is the Euler-Mascheroni constant and
is the Gamma
function (Titchmarsh 1987, Voros 1987). The constant
in the exponent is given by
 |  |  |
(2)
|
 |  |  |
(3)
|
(OEIS A077142). Hadamard used the Weierstrass product theorem to derive this result. The function graph above shows the convergence of the formula along the real axis using the first 100 (red), 500 (yellow), 1000 (green), and 2000 (blue) Riemann zeta function zeros.
The product can also be stated in the alternate form
 |
(4)
|
where 
is the xi-function and
 |
(5)
|
(Havil 2003, p. 204).
The same name is also used in matrix theory for the entrywise Hadamard matrix product.
