Monotonicity results for Dirichlet L-functions

We present some monotonicity results for Dirichlet L-functions associated to real primitive characters. We show in particular that these Dirichlet L-functions are far from being logarithmically completely monotonic. Also, we show that, unlike in the case of the Riemann zeta function, the problem of comparing the signs of dkdsklogL(s,χ) at any two points s1,s2>1 is subtler.