  We investigate the approximation of quadratic Dirichlet $L$-functions over
function fields by truncations of their Euler products. We first establish
representations for such $L$-functions as products over prime polynomials times
products over their zeros. This is the hybrid formula in function fields. We
then prove that partial Euler products are good approximations of an
$L$-function away from its zeros, and that, when the length of the product
tends to infinity, we recover the original $L$-function. We also obtain
explicit expressions for the arguments of quadratic Dirichlet $L$-functions
over function fields and for the arguments of their partial Euler products. In
the second part of the paper we construct, for each quadratic Dirichlet
$L$-function over a function field, an auxiliary function based on the
approximate functional equation that equals the $L$-function on the critical
line. We also construct a parametrized family of approximations of these
auxiliary functions, prove the Riemann hypothesis holds for them, and that
their zeros are related to those of the associated $L$-function. Finally, we
estimate the counting function for the zeros of this family of approximations,
show that these zeros cluster near those of the associated $L$-function, and
that, when the parameter is not too large, almost all the zeros of the
approximations are simple.
