In terms of the gamma functionI1=Γ4(14)4π3I2=3Γ6(13)214/3π4I3=Γ(124)Γ(524)Γ(724)Γ(1124)16√6π3
. There are infinitely many Ramanujan-type formulas for the Ii a few of which are (1+√2)Γ(38)4(2π)5/2√I1=∞∑n=0(2n)!3n!61(−26)n1√2I1=∞∑n=0(2n)!3n!61(−29)n4√3I2=∞∑n=0(2n)!3n!61(28)n1+√2+√6√6I3=∞∑n=0(2n)!3n!61(23(1+√2+√6)3(1+√2))n
There is one other formula that belongs to the family with denominators as powers of 216η(√−7)4=∞∑n=0(2n)!3n!61(212)n
but I don't know if this has an equivalent and analogous integral.
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