Values of zeta functions and class number 1 criterion for the simplest cubic fields

Abstract

Let K Le the simplest cubic field defined by the irreducible polynominal f(x) = x(3) + mx(2) - (m + 3)x +1, where rn is a nonnegative rational integer such that m(2) + 3m + 9 is square-free. We estimate the value of the Dedekind zeta function zeta (K)-(s) at s = -1 and get class number 1 criterion for the simplest cubic fields.