A. F. Lavrik's truncated equations

By a new method, we obtain some known results of A. F. Lavrik (Dokl. Akad. Nauk SSSR, 171, No. 2, 278–280 (1966); Mat. Zametki, 2, No. 5, 475–482 (1967); Izv. Akad. Nauk SSSR, Ser. Mat., 30, No. 2, 433–448 (1966)) regarding the truncated functional equations of various $L$-functions. As an application, we give an estimate of Dedekind's zeta-function of an algebraic number field $K$ of degree $n\leqslant4$ $\zeta_K(\frac12+it)\ll t^{n/6}\log^ct$, $t>1$ and a similar estimate for $L$-series with grцssencharacters. The method of the paper allows us to consider fields of degree $n\leqslant12$.