  We study the monodromy of certain $\ell$-adic local systems attached to the
generalized Kloosterman sheaves constructed by Yun and calculate their Euler
characteristics under standard representations in the cases of symplectic and
split/quasi-split orthogonal groups. This provides evidence for the conjectural
description of their Swan conductors at $\infty$ which is predicted by
Reeder-Yu on the Langlands parameters attached to the epipelagic
representations.
