\begin{definition}[Definition:First Lemniscate Constant]
The '''first lemniscate constant''' is the value of the expression:
{{begin-eqn}}
{{eqn | l = L_1
      | r = \int_0^1 \dfrac {\d x} {\sqrt {1 - x^4} }
      | c = 
}}
{{eqn | r = \dfrac 1 2 \int_0^\pi \dfrac {\d \theta} {\sqrt {1 + \sin^2 \theta} }
      | c = 
}}
{{eqn | r = \dfrac 1 {4 \sqrt {2 \pi} } \paren {\map \Gamma {\dfrac 1 4} }^2
      | c = 
}}
{{eqn | o = \approx
      | r = 1 \cdotp 31102 \, 87771 \, 46059 \, 90523 \ldots
      | c = 
}}
{{end-eqn}}
{{OEIS|A085565}}
\end{definition}