  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.
