$\pi=\varpi_2$ can be considered as a member of the sequence of real numbers $$\varpi_m=2\int\limits_0^1\frac{dt}{\sqrt{1-t^m}},$$ and, for example, the Wallis product formula $$\pi=4\prod_{n=1}^\infty{\frac{2n(2n+2)}{(2n+1)^2}}$$ can be generalized to all these m-clover constants (http://arxiv.org/abs/1212.4178 A Wallis Product on Clovers, by Trevor Hyde): $$\varpi_m = \frac{2(m+2)}{m}\prod_{n=1}^\infty{\frac{2n(2mn+ m +2)} {(2n+1)(2mn+2)}}.$$ On the other hand, for $\pi$ there exists the well known Dalzel's nice integral formula $$\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 \frac{dx}{1+x^2},$$ (see the discussion: Source and context of $\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 dx/(1+x^2)$? ).

I'm curious whether something like the Dalzel's formula exists, for example, for the lemniscate constant $\varpi_4$.