Self-answer: yes, the answer is Proposition 1.12 in Bennett-Martin-O'Bryant-Rechnitzer: https://projecteuclid.org/euclid.ijm/1552442669

They show that, for $\chi$ a Dirichlet character modulo $q\geq 10^5$, the constant term in the Laurent expansion of $L'(s,\chi)/L(s,\chi)$ at $s=0$ has absolute value $\leq 0.2515 q \log q$.

Two quick comments:

• the case $q<10^5$ can be done computationally (is it out there somewhere?)\
• for $q>1$, that constant term equals $\log(2\pi/q) + \gamma - L'(1,\chi)/L(1,\chi)$, by the functional equation.

Their proof (which is based, not to our surprise, on a (good) explicit bound of the "trivial" kind on how close a zero can be to $s=1$) actually gives a lower bound of $\leq c \sqrt{q} \log q$, with $c$ close to $1/40$, under the additional assumption that $\chi$ is primitive.

Self-answer: yes, the answer is Proposition 1.12 in Bennett-Martin-O'Bryant-Rechnitzer: https://projecteuclid.org/euclid.ijm/1552442669

They show that, for $\chi$ a Dirichlet character modulo $q\geq 10^5$, the constant term in the Laurent expansion of $L'(s,\chi)/L(s,\chi)$ at $s=0$ has absolute value $\leq 0.2515 q \log q$.

Two quick comments:

• the case $q<10^5$ can be done computationally (is it out there somewhere?)\
• for $q>1$, that constant term equals $\log(2\pi/q) + \gamma - L'(1,\chi)/L(1,\chi)$, by the functional equation.

Their proof (which is based, not to our surprise, on a (good) explicit bound of the "trivial" kind on how close a zero can be to $s=1$) actually gives an upper bound of $\leq c \sqrt{q} \log q$, with $c$ close to $1/40$, under the additional assumption that $\chi$ is primitive.