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 d k d s k log L ( s , χ ) at any two points s 1 , s 2 > 1 is subtler.