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 $F_1$ $312$ by $600$, in $F_2$ $40$ by $104$, and in $F_3$ $25$ by $51$.