TITLE: Looking for a closed form for a ${}_4 F_3\left(\ldots,1\right)$
QUESTION [19 upvotes]: This question originates from this recent question of Paramanand Singh about a series computed by Ramanujan, probably related to elliptic integrals and Legendre functions.

Is there a closed form for
  $$ {}_4 F_3\left(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2};1,1,1;1\right)=\sum_{n\geq 0}\left[\frac{1}{4^n}\binom{2n}{n}\right]^4=\frac{8}{\pi^3}\int_{0}^{\frac{1}{2}}\frac{K(m)^2}{\sqrt{m(1-m)}}\,dm $$
  ?

Many proofs of $\sum_{n\geq 0}\left[\frac{1}{4^n}\binom{2n}{n}\right]^3=\frac{\pi}{\Gamma\left(\frac{3}{4}\right)^4}$ are well-known, for instance through Clausen formula or Fourier-Legendre series expansions (pages 27-28 here). Such methods do not seem to apply smoothly for computing a closed form for the RHS, neither Parseval's identity applied to
$$ \sum_{n\geq 0}\left[\frac{1}{4^n}\binom{2n}{n}\right]^2 e^{ni\theta} = \frac{2}{\pi}\,K(e^{i\theta})$$
where $e^{i\theta}$ is regarded as the elliptic modulus. Suggestions are welcome.

REPLY [5 votes]: This is more a comment than an answer, but I lack the reputation to comment.
In the paper "moments of elliptic integrals and critical L-values" ( arXiv 1303.2259) equation (30) one finds the identity
$_4F_3(1/2,1/2,1/2,1/2;1,1,1;1)=\frac {16}{\pi^2}L(f,1)$
where $L(f,1)$ is the L-value to the weight four modular form
$f=\eta^4(2\tau)\eta^4(4\tau)$
This L-value was studied by Zagier with the result that it is equal to the value of the apery numbers at -1/2:
$L(f,1)=A_{-1/2}$
Now if one wants to find an expression for the Apery numbers wolfram mathworld gives back the original $_4F_3(...;1)$ , so this gives only different representations.
Moreover, in the Wan paper they state that there exist expressions in terms of gamma functions for odd weight critical L-values, but that these do not exist for the even weights, or at least are unknown. As the L-values corresponding to the $_4F_3(...;1)$ have weight four such expressions are unlikely to exist.