Nonvanishing for cubic <i>L</i>-functions

Abstract We prove that there is a positive proportion of L -functions associated to cubic characters over $\mathbb F_q[T]$ do not vanish at the critical point $s=1/2$ . This achieved by computing first mollified moment using techniques previously developed authors in their work on -functions, and obtaining sharp upper bound for second moment, building Lester Radziwi??, which turn develops further ideas from Soundararajan, Harper Radziwi??. non-Kummer setting when $q\equiv 2 \,(\mathrm {mod}\,3)$ , but our results could be translated into Kummer 1\,(\mathrm as well number-field case (assuming generalised Riemann hypothesis). Our nonvanishing explicit, extremely small, due fact implied constant very large.