To investigate the quality of heavy-traffic approximations for queues with many servers, we consider the steady-state number of waiting customers in an M/D/s queue as s¿8. In the Halfin-Whitt regime, it is well known that this random variable converges to the supremum of a Gaussian random walk. This paper develops methods that yield more accurate results in terms of series expansions and inequalities for the probability of an empty queue, and the mean and variance of the queue length distribution. This quantifies the relationship between the limiting system and the queue with a small or moderate number of servers. The main idea is to view the M/D/s queue through the prism of the Gaussian random walk: as for the standard Gaussian random walk, we provide scalable series expansions involving terms that include the Riemann zeta function.