  The celebrated Riemann-Siegel formula compares the Riemann zeta function on
the critical line with its partial sums, expressing the difference between them
as an expansion in terms of decreasing powers of the imaginary variable $t$.
Siegel anticipated that this formula could be generalized to include the
Hardy-Littlewood approximate functional equation, valid in any vertical strip.
We give this generalization for the first time. The asymptotics contain Mordell
integrals and an interesting new family of polynomials.
