The question below comes to my mind when I am trying to explore something related to the formulas found by [Jesus Guillera][1]:

a)Generalized hypergeometric function 

$${}_3 F_2\left(\begin{matrix}1/4& 3/4& 1/2\\1& 1\end{matrix};\cdots\right)$$

is related to the Dwork family

$x_1^4+x_2^4+x_3^4+x_3^4=4\lambda x_1x_2x_3x_4$ through Picard-Fuchs equation;

b)Certain $\lambda$ corresponds to singular K3 surfaces, which has been given in the note of [N. Elkies and M. Schutt][2]. They calculated all possible rational $\mu,\mu=\lambda^4$ that the corresponding surface is a singular K3 surface(with highest possible Picard number 20), and almost all the numerical values appear in [Ramanujan's 1914 paper][3] on $1/\pi$;

c)J. Guillera has already given certain formulas(formula (92)(93)(94) in W. Zudilin's [paper][4]) on $1/\pi^2$ related to

$${}_5 F_4\left(\begin{matrix}1/6& 2/6& 3/6& 4/6& 5/6\\1& 1&1&1\end{matrix};\cdots\right)$$

and their Dwork family should be

$x_1^6+x_2^6+x_3^6+x_4^6+x_5^6+x_6^6=6\lambda x_1x_2x_3x_4x_5x_6$. Similarly, certain rational $\mu,\mu=\lambda^6$ correspond to Guillera's formulas;

**Question:** It might be a vague question, but is there any **"singular"** member in the family $$x_1^6+x_2^6+x_3^6+x_4^6+x_5^6+x_6^6=6\lambda x_1x_2x_3x_4x_5x_6$$ having similar properties as whose analog is treated in the note of [N. Elkies and M. Schutt][2]? It is said that singular K3 surfaces share the same properties with elliptic curves with complex multiplication. Is there an analog of "complex multiplication" for $$x_1^6+x_2^6+x_3^6+x_4^6+x_5^6+x_6^6=6\lambda x_1x_2x_3x_4x_5x_6?$$ 


  [1]: https://sites.google.com/site/guilleramath/
  [2]: http://www2.iag.uni-hannover.de/~schuett/K3-fam.pdf
  [3]: http://ramanujan.sirinudi.org/Volumes/published/ram06.pdf
  [4]: http://wain.mi.ras.ru/PS/ahs-eng.pdf