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I'm rewriting the whole question in a general form, since that's probably easier to answer and it's also easier to spot the actual question. Assume that we have some finite extension $K/F$ of number fields and assume that the extension is not Galois. Denote the Galois closure by $E/F$. My question is the following:

Is it possible to factor $\zeta_K(s)$ in terms of either Artin $L$-functions of the form $L(s,\chi,E/F)$ or $L$-functions corresponding to intermediate Galois extensions of $E/F$?

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Yes. The first thing to notice is that the Zeta function is the Artin L-function associated to the trivial representation of $Gal(E/K)$ i.e.

$$\zeta_K(s) = L(s, \mathbb{C}, E/K),$$

where $\mathbb{C}$ is endowed with the trivial action of $\mathrm{Gal}(E/K).$

Let $G = \mathrm{Gal}(E/F)$ and $H = \mathrm{Gal}(E/K).$ Now recall that the $L$-function attached to any representation of $\rho$ of $H$ is equal to the $L$-function associated to the induced representation $\rho^G_H$ of $G.$

In particular,

$$L(s, \mathbb{C}, E/K) = L(s, \mathbb{C}[G]\otimes_{\mathbb{C}[H]}\mathbb{C}, E/F).$$

Therefore, if $\chi_1,...,\chi_n$ are the irreducible characters of $G$ and $\chi$ is the character of $\mathbb{C}[G]\otimes_{\mathbb{C}[H]}\mathbb{C},$

$$L(s, \mathbb{C}[G]\otimes_{\mathbb{C}[H]}\mathbb{C}, E/F) =L(s, \chi,E/F) = \displaystyle\prod_{i=1}^nL(s, \chi_i,E/F)^{\langle \chi_i, \chi \rangle},$$

And so,

$$\zeta_K(s) = \displaystyle\prod_{i=1}^nL(s, \chi_i,E/F)^{\langle \chi_i, \chi \rangle}.$$

In the case $H$ and $G$ are explicit, the characters $\chi_1,...,\chi_n$ and $\chi$ are easily calculated and thus give an explicit factorization of $\zeta_K(s).$

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