Family of Approximations for Dirichlet $L$-functions

We introduce an infinite family of approximations for a Dirichlet $L$-function $L(s, \chi)$ arising from truncated Euler products. These approximations are entire functions and satisfy the same functional equation as $L(s, \chi)$. We provide numerical evidence of the accuracy of estimating values of $L(s, \chi)$ in the critical strip using these approximations. We then provide a precise expression for the error of the approximations and show that the error has exponential decay in largest prime considered in the truncated Euler product.