It is known 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][1] 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,
$$L(V,s)=\prod_v L_v(V,s),\qquad\Re(s)\geq 1.$$


  [1]: https://doi.org/10.1080/00029890.1980.11995126