Consider a simple non-compact algebraic group, over any locally compact non-discrete field, which has Kazhdan’s property $\left(T\right)$. For any such group, $G$, we present a Kazhdan set of two elements, and compute its best Kazhdan constant. Then, settling a question raised by Serre and by de la Harpe and Valette, explicit Kazhdan constants for every lattice $\Gamma$ in $G$ are obtained, for a “geometric” generating set of the form $\Gamma \cap B_r$, where $B_r\subset G$ is a ball of radius $r$, and the dependence of $r$ on $\Gamma$ is described explicitly....