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 $\frac{d^k}{ds^k}\log L(s,\chi)$ at any two points $s_1, s_2>1$ is more subtle.