  The classical Kloosterman sums give rise to a Galois representation of the
function field unramfied outside 0 and $\infty$. We study the local monodromy
of this representation at $\infty$ using $l$-adic method based on the work of
Deligne and Katz. As an application, we determine the degrees and the bad
factors of the $L$-functions of the symmetric products of the above
representation. Our results generalize some results of Robba obtained through
$p$-adic method.

Number Theory;Algebraic Geometry