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?