Let $\chi$ be a primitive Dirichlet character of modulus $q>1$. Write, as is customary, $B(\chi)$ for the constant in the expression $$\frac{\Lambda'(s,\chi)}{\Lambda(s,\chi)} = B(\chi) + \sum_\rho \left(\frac{1}{s-\rho} + \frac{1}{\rho}\right),$$ where $\Lambda(s,\chi)$ is a completed Dirichlet $L$-function and $\sum_\rho$ is a sum over its zeros. Obviously, $B(\chi) = \Lambda'(0,\chi)/\Lambda(0,\chi)$. Since $$\frac{L'(s,\chi)}{L(s,\chi)} = \frac{\Lambda'(s,\chi)}{\Lambda(s,\chi)} - \frac{1}{2} \frac{\Gamma'((s+\kappa)/2)}{\Gamma((s+\kappa)/2)} - \frac{1}{2} \log \frac{q}{\pi},$$ where $\kappa = [\chi(-1)=-1]$, we see that $$B(\chi) = b(\chi) - \frac{\gamma}{2} - \kappa \log 2 + \frac{1}{2} \log \frac{q}{\pi},$$ where $b(\chi)$ is the constant term in the Laurent expansion of $L'(s,\chi)/L(s,\chi)$ around $s=0$. We can easily show that $$b(\chi) = \log \frac{2\pi}{q} + \gamma - \frac{L'(1,\overline{\chi})}{L(1,\overline{\chi})}$$ by taking logarithms on both sides of the functional equation. We thus obtain that $$B(\chi) = \frac{1}{2} \log \frac{4^{1-\kappa} \pi}{q} + \frac{\gamma}{2} - \frac{L'(1,\overline{\chi})}{L(1,\overline{\chi})}.$$

It seems clear to me that this expression for $B(\chi)$ must be (very) classical. Now, looking in Montgomery-Vaughan for something else, I see that, in section 10.3, it states that "The constant $B(\chi)$... was long considered to be mysterious; the simple formula (10.39) for it [namely, the expression for $B(\chi)$ right here] is due to Vorhauer (2006)." Here Vorhauer (2006) is an unpublished preprint (not accessible online). I'd gladly give credit where credit is due, but I can't help thinking that this expression must have been known long before 2006. Does anybody have an earlier reference?

(And what would be so mysterious about $B(\chi)$? IMHO, it is just tricky for the same reason that $L'(1,\chi)/L(1,\chi)$, viz., the possibility of a Siegel zero. Or is it just that we don't have an expression for it as nice as the class number formula? (Do we? EDIT: for $\chi$ odd, we do; see Prop. 10.3.5 (due to...?) in Henri Cohen's Number Theory.) On the issue of bounding it, see $|L'(1,\chi)/L(1,\chi)|$.)

1918, page 216, in the case $q=4$ (which is the case that Landau needed in that paper). The proof for arbitrary $q$ is the same. $\endgroup$ – H A Helfgott Oct 8 '19 at 15:54The Euler-Kronecker constant, Math. Notes, vol. 87, 2010), using Stechkin's idea. $\endgroup$ – Vesselin Dimitrov Oct 13 '19 at 2:07