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$12$ in $\mathrm{PSL}_2(\mathcal{O}_K)$ CITE{Chu2020}, so twelve times HREF{#-3}[the $D=-3$ entry] is HREF{Hyperbolic_volumes_of_the_prime_knots_with_at_most_ten_crossings#4,1}[the- volume of the figure-eight knot complement]. The HREF{#-4}[$D=-4$ entry] is HREF{Values_of_the_Clausen_functions_at_rational_multiples_of#2,1/2}[Catalan's+ volume of the figure-eight knot complement]. The HREF{#-4}[$D=-4$ entry] is HREF{Values_of_the_Clausen_functions_at_rational_multiples_of_pi#2,1/2}[Catalan's constant] divided by $3$. Formulas:
Dirichlet $L$-function], the same volume is $|D|^{3/2}L(2,\chi_D)/24$. formula-clausen: $\mathrm{Vol}\bigl(\mathrm{PSL}_2(\mathcal{O}_K)\backslash\mathbb{H}^3\bigr)- = |D|\sum_{a=1}^{|D|-1}\chi_D(a)\mathrm{Cl}_2(2\pi a/|D|)/24$, where HREF{Values_of_the_Clausen_functions_at_rational_multiples_of}[$\mathrm{Cl}_2$]+ = |D|\sum_{a=1}^{|D|-1}\chi_D(a)\mathrm{Cl}_2(2\pi a/|D|)/24$, where HREF{Values_of_the_Clausen_functions_at_rational_multiples_of_pi}[$\mathrm{Cl}_2$] is the Clausen function. Similar tables:
relation: $\kappa_D$, the residue at $s=1$ of the same $\zeta_K$, under the same $D$-- table: HREF{Values_of_the_Clausen_functions_at_rational_multiples_of}[Values of- the Clausen functions at rational multiples of $\pi$]+- table: HREF{Values_of_the_Clausen_functions_at_rational_multiples_of_pi}[Values+ of the Clausen functions at rational multiples of $\pi$] relation: the finite Clausen sum in CITE{formula-clausen} Links:
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