Local approximations to the exchange-correlation functional are of special interest because of their cost advantages and their useful accuracy for efficient calculations on systems (such as many transition metal catalysts) with significant multiconfigurational wave function character. We present a meta-GGA exchange-correlation functional, called M11-L, that employs dual-range local exchange to provide broad accuracy for both single-configurational and multiconfigurational molecules and for solid-state lattice constants. Also notable is the high accuracy (for a local functional) for chemical reaction barrier heights. The mean unsigned error on a broad chemistry database of 338 energetic data is lower than that for any other known functional, even hybrid functionals and range-separated hybrid functionals. This success shows that the dependence of the exchange energy density on interelectronic distance is quite different at short-range and long-range, and it establishes a new standard for the limit of what can be achieved with a local exchange-correlation functional.