  We report the computation of a family of traveling wave solutions of pipe
flow up to $Re=75000$. As in all lower-branch solutions, streaks and rolls
feature prominently in these solutions. For large $Re$, these solutions develop
a critical layer away from the wall. Although the solutions are linearly
unstable, the two unstable eigenvalues approach 0 as $Re\to\infty$ at rates
given by $Re^{-0.41}$ and $Re^{-0.87}$ -- surprisingly, the solutions become
more stable as the flow becomes less viscous. The formation of the critical
layer and other aspects of the $Re\to\infty$ limit could be universal to
lower-branch solutions of shear flows. We give implementation details of the
GMRES-hookstep and Arnoldi iterations used for computing these solutions and
their spectra, while pointing out the new aspects of our method.
