Hadamard's infinite product is one expansion of it, and it can also be expanded into a Laurent series using Stieltjets constants.  
Hadamard's infinite product is one expansion of it, and it can also be expanded into a Laurent series using Stieltjets constants.  The Dirichlet L-function with chi as a trivial character is equivalent to it, and its value at 3 is known as Apery's constant.  Work by Ramanujan showed that its value at -1 is equal to -1/12, and his namesake series for it can also show that it is zero at any negative even number.  Hardy showed that it has an infinite number of roots with real part ½, but it is not yet known whether all of its roots lie on that critical line.  FTP, name this function intimately related to the prime number theorem and the subject of a namesake unsolved hypothesis.
Hadamard's infinite product is one expansion of it, and it can also be expanded into a Laurent series using Stieltjets constants.  
{Riemann zeta} function