diff --git a/crates/ekore/src/harmonics/cache.rs b/crates/ekore/src/harmonics/cache.rs
index 89ae8f3e0..6d88b46cf 100644
--- a/crates/ekore/src/harmonics/cache.rs
+++ b/crates/ekore/src/harmonics/cache.rs
@@ -38,6 +38,10 @@ pub enum K {
     Sm2e,
     /// $S_{-2}(N)$ odd moments
     Sm2o,
+    /// $S_{-3}(N)$ even moments
+    Sm3e,
+    /// $S_{-3}(N)$ odd moments
+    Sm3o,
     /// $S_{-2,1}(N)$ even moments
     Sm21e,
     /// $S_{-2,1}(N)$ odd moments
@@ -90,6 +94,8 @@ impl Cache {
             K::Sm1o => w1::Sm1o(self.get(K::S1), self.get(K::S1mh)),
             K::Sm2e => w2::Sm2e(self.get(K::S2), self.get(K::S2h)),
             K::Sm2o => w2::Sm2o(self.get(K::S2), self.get(K::S2mh)),
+            K::Sm3e => w3::Sm3e(self.get(K::S3), self.get(K::S3h)),
+            K::Sm3o => w3::Sm3o(self.get(K::S3), self.get(K::S3mh)),
             K::Sm21e => w3::Sm21e(self.n, self.get(K::S1), self.get(K::Sm1e)),
             K::Sm21o => w3::Sm21o(self.n, self.get(K::S1), self.get(K::Sm1o)),
         };
diff --git a/crates/ekore/src/harmonics/w3.rs b/crates/ekore/src/harmonics/w3.rs
index 2987db7ff..cbcb5df59 100644
--- a/crates/ekore/src/harmonics/w3.rs
+++ b/crates/ekore/src/harmonics/w3.rs
@@ -1,5 +1,6 @@
 //! Harmonic sums of weight 3.
 use num::complex::Complex;
+use num::traits::Pow;
 use std::f64::consts::LN_2;
 
 use crate::constants::{ZETA2, ZETA3};
@@ -14,6 +15,16 @@ pub fn S3(N: Complex<f64>) -> Complex<f64> {
     0.5 * cern_polygamma(N + 1.0, 2) + ZETA3
 }
 
+/// Analytic continuation of harmonic sum $S_{-3}(N)$ for even moments.
+pub fn Sm3e(hS3: Complex<f64>, hS3h: Complex<f64>) -> Complex<f64> {
+    1. / (2_f64).pow(2) * hS3h - hS3
+}
+
+/// Analytic continuation of harmonic sum $S_{-3}(N)$ for odd moments.
+pub fn Sm3o(hS3: Complex<f64>, hS3mh: Complex<f64>) -> Complex<f64> {
+    1. / (2_f64).pow(2) * hS3mh - hS3
+}
+
 /// Analytic continuation of harmonic sum $S_{-2,1}(N)$ for even moments.
 pub fn Sm21e(N: Complex<f64>, hS1: Complex<f64>, hSm1: Complex<f64>) -> Complex<f64> {
     let eta = 1.;