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An expression for the hyperbolic volume of the Weeks manifold is given on Wikipedia: $$\frac{3\cdot23^{3/2}\zeta_K(2)}{4\pi^4}=0.94270736277\dots\tag1$$ Here $\zeta_K$ is the Dedekind zeta function corresponding to the number field $t^3-t+1$. This can be easily evaluated in PARI/GP:

? 3*23^(3/2)*lfun(x^3-x+1,2)/(4*Pi^4)
%8 = 0.94270736277692772092129960309221164758

When I edited the corresponding OEIS entry A126774 in 2016 to include the 2009 proof that the Weeks manifold minimises hyperbolic volume among closed orientable hyperbolic $3$-manifolds, there was already a comment giving an alternative formula for the volume from a Herman Jamke: $$\operatorname{Im}(\operatorname{Li}_2(t_0)-\ln|t_0|\ln(1-t_0))=0.94270736277\dots\tag2$$ Here $t_0$ is the root with positive imaginary part of $t^3-t+1$. This is also easily evaluable in PARI/GP – and mpmath and Mathematica and other software since there is no Dedekind zeta function involved:

? th = polroots(x^3-x+1)[3];
? imag(dilog(th) + log(abs(th))*log(1-th))
%2 = 0.94270736277692772092129960309221164759

No reference whatsoever had been provided for the equivalence between these two expressions in the nine years between Jamke's comment and my OEIS edit, and there is still none. While I simply shrugged and moved on back then, I have gotten a bit into zeta and $L$-functions from needing to compute the Dirichlet/analytic density in Chebotarev's density theorem (cf. here, which uses the not-always-defined natural density).

How can $(2)$ be derived from $(1)$?

Parcly Taxel
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    Related: The article D.Zagier "The Dilogarithm Function" in "Frontiers in Number Theory, Physics, and Geometry." II pp. 3-65. However, one needs extra work to connect it to the OP since the zeta-function in Zagier's paper is associated with a quadratic extension of ${\mathbb Q}$ (because the hyperbolic orbifolds in Zagier's paper are the ones associated with Bianchi groups), while the zeta-function in OP is for a cubic extension of ${\mathbb Q}$. – Moishe Kohan May 31 '21 at 02:18

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This (and something much more general) is briefly explained in I.5 of The Dilogarithm Function, D Zagier,Frontiers in Number Theory, Physics, and Geometry II, 2007 (link below), particularly on page 18.

A full explanation would be too long to fit with the MSE format, but briefly, both expressions come from computing the volume in two different ways (one exploiting the description in terms of an arithmetic manifold and the other by ultimately a decomposition into ideal tetrahedra). I.4 and I.5 discuss exactly the relation between the volumes of hyperbolic manifolds and Dedekind zeta functions. The paper clearly explains and gives references to the precise relationship between values of Dedekind zeta functions at 2 and values of dilogarithms.

Zagier, Don, The dilogarithm function, Cartier, Pierre (ed.) et al., Frontiers in number theory, physics, and geometry II. On conformal field theories, discrete groups and renormalization. Papers from the meeting, Les Houches, France, March 9–21, 2003. Berlin: Springer (ISBN 978-3-540-30307-7/hbk). 3-65 (2007). ZBL1176.11026.

The paper is also (at time of writing) also available on the author's website here.

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