Counting primes in residue classes

We explain how the Meissel-Lehmer-Lagarias-Miller-Odlyzko method for computing $\pi(x)$ can be used for computing efficiently $\pi(x,k,l)$, the number of primes congruent to $l$ modulo $k$ up to $x$. As an application, we computed the number of prime numbers of the form $4n \pm 1$ less than $x$ for several values of $x$ up to $10^{20}$ and found a new region where $\pi(x,4,3)$ is less than $\pi(x,4,1)$ near $x = 10^{18}$.