In this paper, we prove that the multiple $L$-value of conductor $4$ can be expressed in terms of lower depth multiple $L$-values under the condition on the parity of its depth and weight. This can be regarded as a character analogue of what is called the "parity result" for multiple zeta values which was proved by Zagier.
"On the Parity Conjecture for Multiple $L$-values of Conductor Four." Tokyo J. Math. 30 (1) 21 - 40, June 2007. https://doi.org/10.3836/tjm/1184963645