Divisible formal weight enumerators and extremal polynomials not satisfying the Riemann hypothesis

In this paper, first we formulate the notion of divisible formal weight enumerators and propose an algorithm for the efficient search of the formal weight enumerators divisible by two. The main tools are the binomial moments. It leads to the discovery of several new families of formal weight enumerators. Then, as a result, we find examples of extremal formal weight enumerators which do not satisfy the Riemann hypothesis.