**I. Introduction**

In *Identities Inspired from Ramanujan’s Notebooks*, Simon Plouffe recounts how, based on Ramanujan’s,

he found,

and similar ones for other *s* = 4m+3. On a hunch, and using Mathematica’s *LatticeReduce* function, I found that,

etc.

**II. Functions**

If we define,

then Plouffe discovered integer relations between,

for odd *s*, with *s* = 3 being,

Eliminating leads to the 3-term equalities in the *Introduction*. See Chamberland’s and Lopatto’s *Formulas for Odd Zeta Values*. On the other hand, by defining the function,

I observed integer relations between,

also for odd *s*, with *s* = 3 as,

and so on. Eliminating leads to the 4-term equalities in the *Introduction*.

**III. Conjecture**

The 4-term equalities have coefficients that are simple except for one term. Recall that,

*Conjecture*:

*“Using the positive case of* *for s = 4m+3, and the negative for s = 4m+5, then in the equation,*

*is a rational number.”*

The first few for *s* = {3, 7, 11,…} are while for *s* = {5, 9, 13,…} are These rationals may have a closed-form expression in terms of *Bernoulli numbers*, but I do not yet know the exact formulation.