Explicit upper bounds for 𝐿-functions on the critical line

We find an explicit upper bound for general L L -functions on the critical line, assuming the Generalized Riemann Hypothesis, and give as illustrative examples its application to some families of L L -functions and Dedekind zeta functions. Further, this upper bound is used to obtain lower bounds beyond which all eligible integers are represented by Ramanujan’s ternary form and Kaplansky’s ternary forms. This improves on previous work by Ono and Soundararajan on Ramanujan’s form and by Reinke on Kaplansky’s forms with a substantially easier proof.