Is the $0$-error capacity of $7$-cycle:


$(1)$ known to be of form $7^q$ for some $q\in \mathbb Q$?



Form of the Shannon capacity for Heptagon?
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). 


Please see


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.


