On a factorization of Riemann's $ζ$ function with respect to a quadratic field and its computation

Let $K$ be a quadratic field, and let $ζ_K$ its Dedekind zeta function. In this paper we introduce a factorization of $ζ_K$ into two functions, $L_1$ and $L_2$, defined as partial Euler products of $ζ_K$, which lead to a factorization of Riemann's $ζ$ function into two functions, $p_1$ and $p_2$. We prove that these functions satisfy a functional equation which has a unique solution, and we give series of very fast convergence to them. Moreover, when $Δ_K>0$ the general term of these series at even positive integers is calculated explicitly in terms of generalized Bernoulli numbers.

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