Let $H^2_m$ be the Hilbert function space on the unit ball in $C{m}$ defined by the kernel $k(z,w) = (1-langle z,w rangle)^{-1}$. For any weak zero set of the multiplier algebra of $H^2_m$, we study a natural extremal function, $E$. We investigate the properties of $E$ and show for example that $E$ tends to $0$ at almost every boundary point. We also give several explicit examples of the extremal function and compare the behaviour of $E$ to the behaviour of $delta^*$ and $g$, the corresponding extremal function for $H^infty$ and the pluricomplex Green function, respectively.