  We propose an investigation on the Northcott, Bogomolov and Lehmer properties
for special values of L-functions. We first introduce an axiomatic approach to
these three properties. We then focus on the Northcott property for special
values of L-functions. In the case of L-functions of pure motives, we prove a
Northcott property for special values located at the left of the critical
strip, assuming that the L-functions in question satisfy some expected
properties. Inside the critical strip, focusing on the Dedekind zeta function
of number fields, we prove that such a property does not hold for the special
value at one, but holds for the special value at zero, and we give a related
quantitative estimate in this case.
