This is not an answer.

As was noted in the comments, the function $C(x)$ can be expressed in terms of the Gamma function, see Corollary 1 on page 145 in Ramanujan's Notebook II which attributes this result to G. Bauer: Von einem Kettenbruche Euler's und einem Theorem von Wallis, Abh. Bayer. Akad. Wiss. 11(1872), 96 - 116 and to the first Ramanujan letter to Hardy.

$$
C(x) = \frac{\Gamma^2(\frac{x+1}{4})}{\Gamma^2(\frac{x+3}{4})}
$$

The AGM function can be expressed in terms of complete elliptic integral of the first kind, as is noted on the Wikipedia page, which is given by Gauss hypergeometric function ${}_2F_1(-\frac{1}{2}, \frac{1}{2}; 1; k^2)$ whereas the function $C(x)$ is given by ${}_2F_1(-\frac{1}{2}, -\frac{1}{2};x,1)$.

There is certainly more to the story, see related question on math.stackexchange and equation (99) at MathWorld.

The explicit relation connecting Gamma values at certain rational points and elliptic integral can be found e.g. in the article of Borwein and Zucker, book Pi and the AGM by Borwein, Borwein and a more recent article by Vidunas.