Let $d$ be a fundamental discriminant and let $\chi$ be the associated primitive real character of modulus $\vert d \vert$. Assuming GRH, Littlewood proved that as $\vert d \vert$ grows large,
$$L(1, \chi) \leq (2 + o(1)) e^\gamma \log \log(\vert d \vert).$$


Granville and Soundararajan provide a treasure-trove of information about $L(1,\chi)$ in their GAFA paper (Vol. 13, 2003).


But I've never seen a bound as above in which the $o(1)$ is made explicit. (I admit that when logarithms are inside of logarithms, my brain tries to jump out of my ear and run away, so I've only looked for an hour or two.) Is there any statement proven (assuming GRH) along the lines of the following -- for some explicit constant $K$ and explicit decaying function $F$?


Desired form: If $\vert d \vert > K$ then $L(1, \chi) \leq (2 + F(\vert d \vert)) e^\gamma \log \log(\vert d \vert).$


If not, is there a well-known obstruction to proving such statements?



Effective bound of $L(1,\chi)$
Explicit upper and lower bounds for $L(1,\chi)$, conditional on the generalised Riemann hypothesis, are given in Theorem 1.5 of the paper ["Conditional bounds for the least quadratic non-residue and related problems"](https://doi.org/10.1090/S0025-5718-2015-02925-1) by Youness Lamzouri, Xiannan Li and Kannan Soundararajan. In particular, for $q \geq 10^{10}$ and $\chi$ a primitive Dirichlet character modulo $q$, we have the explicit upper and lower bounds
$$|L(1,\chi)| \leq 2e^{\gamma}\left(\log \log q - \log 2 + \frac{1}{2} + \frac{1}{\log \log q}\right)$$
and
$$\frac{1}{|L(1,\chi)|} \leq \frac{12e^{\gamma}}{\pi^2} \left(\log \log q - \log 2 + \frac{1}{2} + \frac{1}{\log \log q} + \frac{14 \log \log q}{\log q}\right).$$
(There is a corrigendum to this paper, but Theorem 1.5 remains unchanged.)


This method generalises quite nicely to other $L$-functions, as explained in the paper ["Explicit bounds for $L$-functions on the edge of the critical strip"](https://doi.org/10.1016/j.jnt.2018.01.001) by Allysa Lumley.


