  The formal weight enumerators were first introduced by M. Ozeki. They form a
ring of invariant polynomials which is similar to that of the weight
enumerators of Type II codes. Later, the zeta functions for linear codes were
discovered and their theory was developed by I. Duursma. It was generalized to
certain invariant polynomials including Ozeki's formal weight enumerators by
the present author. One of the famous and important problems is whether
extremal weight enumerators satisfy 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.
