## Xi Function

 (1)

where is the Riemann Zeta Function and is the Gamma Function (Gradshteyn and Ryzhik 1980, p. 1076). The function satisfies the identity
 (2)

The zeros of and of its Derivatives are all located on the Critical Strip , where . Therefore, the nontrivial zeros of the Riemann Zeta Function exactly correspond to those of . The function is related to what Gradshteyn and Ryzhik (1980, p. 1074) call by
 (3)

where . This function can also be defined as
 (4)

giving
 (5)

The de Bruijn-Newman Constant is defined in terms of the function.