Given a one-dimensional Lévy process \((X_t)_{t\geq0}\), the cumulative distribution function of the supremum process \(M_t=\sup_{0\leq s\leq t}X_s\) is studied. Relying on fluctuation theory, simple sharp bounds for \(\operatorname P(M_t<x)\) are derived for a large class of Lévy processes. Recalling that the ascending ladder-time process is given as the right-continuous inverse of the local time, it is proved under mild assumptions that \[ \operatorname P(M_t<x)\approx\min(1,\kappa(1/t,0)V(x)),\quad t,x>0, \] where \(V(x)\) and \(\kappa(z,0)\) are the renewal function for the ascending ladder-height process, and the Laplace exponent of the ascending ladder-time process corresponding to \(X_t\), respectively, and the equality holds up to constants. Under symmetry and some regularity of the characteristic exponent \(\Psi\) of \(X_t\), the above formula takes the very simple form \[ \operatorname P(M_t<x)\approx\min(1,(t\Psi(1/x))^{-1/2}),\quad t,x>0. \] Moreover, an integral representation of the Laplace transform of the distribution function of \(M_t\) is proved if \(X_t\) is symmetric and \(\Psi\) is increasing on the positive half line.