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The usual (?) proof of Chebotarev's density theorem yields that $L(V,s)$ is analytic and non-vanishing in $\Re(s)\geq 1$. Equivalently, $\log L(V,s)$ is analytic in $\Re(s)\geq 1$. Note that $\log L(V,s)$ is given by an absolutely convergent Dirichlet series in $\Re(s)>1$. This Dirichlet series has bounded coefficients, it is supported on prime powers, and the contribution of a given prime $v$ equals $\log L_v(V,s)$. In particular, $$\log L(V,s)=\sum_v\log L_v(V,s),\qquad\Re(s)>1.$$ By Ingham's Tauberian theorem (for which Newman gave a simple proof in 1980), it follows that the Dirichlet series of $\log L(V,s)$ converges in $\Re(s)\geq 1$. Comparing this Dirichlet series to the Dirichlet series of the various terms $\log L_v(V,s)$, it actually follows that $$\log L(V,s)=\sum_v\log L_v(V,s),\qquad\Re(s)\geq 1.$$ Equivalently, $$L(v,s)=\prod_v L_v(V,s)\qquad\Re(s)\geq 1.$$