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

Abstract

Let $K$ be the simplest cubic field defined by the irreducible polynomial $$f(x) = x^{3}+mx^{2}-(m+3)x+1,$$ where $m$ 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.