On quotients of Riemann zeta values at odd and even integer arguments

We show for even positive integers $n$ that the quotient of the Riemann zeta values $\zeta (n+1)$ and $\zeta (n)$ satisfies the equation $$\frac{\zeta (n+1)}{\zeta (n)}=(1-\frac{1}{n})(1-\frac{1}{2^{n+1}-1})\frac{{\mathcal{L}}^{\star }({\mathfrak{p}}_n)}{{\mathfrak{p}}_n^{\mathrm{\prime }}(0)},$$ where ${\mathfrak{p}}_n\in \mathbb{Z}[x]$ is a certain monic polynomial of degree $n$ and ${\mathcal{L}}^{\star }:\mathbb{C}[x]\to \mathbb{C}$ is a linear functional, which is connected with a special Dirichlet series. There exists the decomposition ${\mathfrak{p}}_n(x)=x(x+1){\mathfrak{q}}_n(x)$. If $n=p+1$ where $p$ is an odd prime, then ${\mathfrak{q}}_n$ is an Eisenstein polynomial and therefore irreducible over $\mathbb{Z}[x]$.