Let \(F\) be a number field with discriminant \(d_ F\) and with \(r_ 1\) real and \(r_ 2\) pairs of complex conjugate embeddings \(\sigma_ j: F\to {\mathbb{C}}\), \(1\leq j\leq r_ 1+r_ 2\). Consider the homomorphism \(\Delta\) : \({\mathbb{Q}}[{\mathbb{P}}^ 1_ F\setminus \{0,1,\infty \}]\to (\wedge^ 2F^{\times}\otimes F^{\times})\otimes {\mathbb{Q}}\) with \(\Delta\) : \(\{\) \(x\}\mapsto (1-x)\wedge x\otimes x\). Also, write \({\mathcal L}_ 3(z)={\mathfrak R}[\text{Li}_ 3(z)-\log | z| \cdot \text{Li}_ 2(z)+(1/3)\log^ 2| z| \cdot \text{Li}_ 1(z)]\) for the ‘cleaned’ trilogarithm function. This function \({\mathcal L}_ 3\) is real-analytic on \({\mathbb{P}}^ 1_{{\mathbb{C}}}\{0,1,\infty \}\) and continuous on all of \({\mathbb{P}}^ 1_{{\mathbb{C}}}\). Then the main result announced in this paper says: There exist \(y_ 1,...,y_{r_ 1+r_ 2}\in \text{Ker}\, \Delta \subset {\mathbb{Q}}[{\mathbb{P}}^ 1_ F\setminus \{0,1,\infty \}]\) such that the Dedekind zeta function \(\zeta_ F(s)\) of \(F\) satisfies: \[ \zeta_ F(3)=\frac{\pi^{3r_ 2}}{\sqrt{| d_ F|}}\cdot \det ({\mathcal L}_ 3(\sigma_ iy_ i))_{1\leq i,j\leq r_ 1+r_ 2}. \] For \(s=2\) a similar result in terms of the dilogarithm was proved by D. Zagier, and, in fact, similar expressions for \(\zeta_ F(s)\), for all integers \(s\geq 3\), are conjectured by Zagier. Zagier’s conjecture amounts to the construction of higher Bloch groups \({\mathcal B}_ m(F)\) and canonical maps \({\mathcal B}_ m(F)\to K_{2m-1}(F)\) with finite kernels and cokernels such that the composition with Borel’s regulator map \(K_{2m- 1}(F)\otimes {\mathbb{Q}}\to {\mathbb{R}}^{r_ 2}\) or \({\mathbb{R}}^{r_ 1+r_ 2}\), according to whether m is even or odd, coincides with a well-defined expression \(P_ m\) involving the higher polylogarithm functions \(\text{Li}_ 1,...,\text{Li}_ m\). For the value \(\zeta_ F(m)\) one should have an expression similar to the one above, with \({\mathcal L}_ 3\) replaced by \(P_ m\). In this paper a geometric construction of \({\mathcal B}_ 3(F)\) is given and \({\mathcal B}_ 3(F)\otimes {\mathbb{Q}}\) is shown to be the quotient \({\mathbb{Q}}[{\mathbb{P}}^ 1_ F\setminus \{0,1,\infty \}]/R_ 3\), where \(R_ 3\) is an expression determined by suitable configurations of seven points in \({\mathbb{P}}^ 2_ F\). Thus \(R_ 3\) depends on three parameters and it is announced to lead to a functional equation for \({\mathcal L}_ 3\) involving 22 terms. For a special configuration this functional equation reduces to the Spence-Kummer equation. It is conjectured that any functional equation for \({\mathcal L}_ 3\) can be formally deduced from this functional equation together with two obvious ones.