Stupid of me. As O. Gorodetsky mentions, these are classical: $$F_1=(91\zeta(3)-2\pi^3\sqrt{3})/432$$ $$F_2=(28\zeta(3)-\pi^3)/64$$ $$F_3=(117\zeta(3)-2\pi^3\sqrt{3})/243$$
In addition, note that there are almost identical cfracs for the same linear combinations where the $-$ sign is replaced by a $+$ sign: replace in $F_1$ the $312$ by $600$, in $F_2$ the $40$ by $104$, and in $F_3$ the $15$ by $51$.
Added: we have $F_4=(7/16)\zeta(3)=0.525899...$. This is due to Y. Yang and is referred to in my arXiv paper mentioned in the post.
