We have	$$C_2(-17,-6,-72)=-(5/8)L(\chi_{-3},2)$$ and
$$C_2(10,3,-9)=(1/2)L(\chi_{-3},2)$$ so both are	proportional to	what you
call Gieseking's constant but which is simply the value	at 2 of	the
L function of the nontrivial character modulo 3, close analogue	to Catalan's
constant which is the same with	the nontrivial character modulo	4.

All the	other "??" that	you quote, both	in degree 2 and	in degree 3
are divergent cfracs (by the way, "degree" is more proper than "level").

Finally	just a typo: $C_3(12,4,-32)=-(7/32)\zeta(3)$ (minus sign omitted).

Two useful references:

O. Gorodetsky, New representations for all sporadic Ap\'ery-like sequences, with applications to congruences,
arXiv:2102:2102.11839 (2021)

and

Y. Yang, Ap\'ery limits and special values of $L$-functions,
J. Math. Anal. Appl. {\bf 343} (2008), 492--513.