Language is supported by a distributed network of brain regions with a particular contribution from the left hemisphere. A multi-level understanding of this network requires studying the genetic architecture of its functional connectivity and hemispheric asymmetry. We used resting state functional imaging data from 29,681 participants from the UK Biobank to measure functional connectivity between 18 left-hemisphere regions implicated in multimodal sentence-level processing, as well as their homotopic regions in the right-hemisphere, and interhemispheric connections. Multivariate genome-wide association analysis of this total network, based on common genetic variants (with population frequencies above 1%), identified 14 loci associated with network functional connectivity. Three of these loci were also associated with hemispheric differences of intrahemispheric connectivity. Polygenic dispositions to lower language-related abilities, dyslexia and left-handedness were associated with generally reduced leftward asymmetry of functional connectivity, but with some trait- and connection-specific exceptions. Exome-wide association analysis based on rare, protein-altering variants (frequencies < 1%) suggested 7 additional genes. These findings shed new light on the genetic contributions to language network connectivity and its asymmetry based on both common and rare genetic variants, and reveal genetic links to language-related traits and hemispheric dominance for hand preference.