De Bruijn-Newman constant denoted as Λ is closely conected with Riemann hypothesis on zeroes of general Euler - Riemann ζ(s) function.
In briefly it is said that the Riemann hypothesis is equivalent to the conjecture that Λ ≤ 0. De Bruijn 1950 first estimated to be Λ ≤ 1/2 according to Newman's work who first estimated to it would be Λ ≥ 0. Serious calculations on Λ was made since 1988 and are still made as we see from table:
a year lower bound of Λ 1988 -50 1991 -5 1990 -0.385 1994 -4.379 · 10 -6 1993 -5.895 · 10 -9 2000 -2.7 · 10 -9