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.