The main results of this paper concern the Northcott property for special values of \(L\)-functions. First of all, the authors prove a general result about \(L\)-functions associated to pure motives, and their special values at the left of the critical strip. Another case in which the expected properties of \(L\)-functions are known is provided by the Dedekind zeta functions \(\zeta_F(s)\) associated to number fields \(F\). More precisely, let \[ \zeta_F^*(s)=\lim_{t\to s} \frac{\zeta_F(t)}{(t-s)^{\mathrm{ord}_{t=s}(\zeta_F(t))}} \] denote the first nonzero coefficient of the Taylor series for \(\zeta_F(s)\) around \(s\). The complex number \(s\) satisfies the Northcott property for a real positive number \(B\) if the set of isomorphism classes of number fields \(F\) given by \[ S_{B,s}= \lbrace [F]: |\zeta_F^*(s)|\leq B \rbrace \] is finite. It is shown that the Northcott property holds for a negative integer \(n\) or \(n=0\), and it does not hold if \(n\) is a positive integer. They also give a quantitative estimate for the size of the set \(S_{B,s}\) for th integers \(s=n\) such that the Northcott property holds.