Is the $0$-error capacity of $7$-cycle:
$(1)$ known to be of form $7^q$ for some $q\in \mathbb Q$?
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Indeed as Daniel pointed out, the Shannon capacity of $C_7$ is still unknown. Also, to say something closer to your question, its specific form is still unknown (in my knowledge).
T. Bohman, A limit theorem for the Shannon capacity of odd cycles II, Proc. Amer. Math. Soc. 133 (2005), no. 2, 537-543.
or the even more specific
A. Vesel, J. Zerovnik, Improved lower bounds on the Shannon capacity of C_7, Information Processing Letters, Volume 81, Issue 5, 16 March 2002, Pages 277–282.
According to http://www.stahlke.org/dan/publications/phd-thesis.pdf (bottom of page 5) the Shannon capacity of the heptagon is still unknown.