  We investigate whether the six-loop beta function of the $\lambda \phi^4_4$
theory exhibits evidence for an ultraviolet zero. As part of our analysis, we
calculate and analyze Pad\'e approximants to this beta function. Extending our
earlier results at the five-loop level, we find that in the range of $\lambda$
where the perturbative calculation of the six-loop beta function is reliable,
the theory does not exhibit robust evidence for an ultraviolet zero.
