It's all clear now, thanks a lot! Edwards' suggestion was misleading.

Thanks! That sounds good, I'll think about it tomorrow. And my question was about the limit of $I(\beta)$, not about the log. I didn't phrase it well, sorry.

Sure, but for dominated convergence I would have to bound the absolute value of the integrand by an (integrable) function that only depends on $s$ on a region $(a-i\infty,a+i\infty) \times (\sigma+i\tau_0,\sigma+i\infty)$, but the Taylor expansion isn't valid on all of this region. But maybe I misunderstand what Edwards or you are suggesting?