  The classical criterion of Jensen for the Riemann hypothesis is that all of
the associated Jensen polynomials have only real zeros. We find a new version
of this criterion, using linear combinations of Hermite polynomials, and show
that this condition holds in many cases. Detailed asymptotic expansions are
given for the required Taylor coefficients of the xi function at $1/2$ as well
as related quantities. These results build on those in the recent paper of
Griffin, Ono, Rolen and Zagier.
