For p = 1 , it is known that C s , 1 = 2 ζ ζ ( s ) , where ζ ζ ( s ) is the Riemann zeta function. Using a modular form approach to linear programming, Viazovska together with coauthors established in a 2022 paper that in dimensions p = 8 and p = 24 , the values of C s , p , s > p , are given by the Epstein zeta function associated with the E 8 lattice and Leech lattice, respectively. It is conjectured that for p = 2 , the value of C s , p is similarly determined as the value of the Epstein zeta function for the hexagonal lattice. Finally, in every dimension p ≥ ≥ 1 it is known that when s = p , the scaling of E s ( A , N ) becomes N 2 log ⁡ ⁡ N rather than N 2 = N 1 + s / p , and the value of C s , p can be computed explicitly as the volume of the unit p -dimensional ball : C s , p = H p ( B p ) = π π p / 2 Γ Γ ( 1 + p / 2 ) . The following connection between the constant C s , p and the problem of sphere packing is known: lim s → → ∞ ∞ ( C s , p ) 1 / s = 1 s ( α α p Δ Δ p ) 1 / p , where α α p is the volume of a p-ball and Δ Δ p = sup ρ ρ ( P ) , where the supremum is taken over all families P of non-overlapping unit balls such that the limit ρ ρ ( P ) = lim r → → ∞ ∞ λ λ ( [ − − r , r ] p ∩ ∩ ⋃ ⋃ B ∈ ∈ P B ) ( 2 r ) p exists.