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The question is in the title : suppose RH is false. Would it imply that there exists a yet to be discovered functional equation relating the values of the Riemann zeta function at different points themselves related in some specific way ? As far as I know, the functional equation as we know it is required as a necessary condition for RH to be true, but not a sufficient one. Are references on this subject available ?

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Suppose RH is false. Would this imply that there exists a yet to be discovered functional equation?

No.

As far as I know, the functional equation is a necessary condition for RH to be true, but not sufficient.

This is an odd statement. There are plenty of functions that have constrained zeroes but no functional equation. So this is a bit of a mystery.

However, you may be interested to know that there other functions $L(s)$ which are similar to the Riemann zeta function: $L(s) = \sum a(n) n^{-s}$ and $L(s)$ has a functional equation of the shape $L(s) \sim L(1-s)$, just like $\zeta(s)$. But some of these functions have "nontrivial" zeroes off the line $\mathrm{Re} (s) = \frac{1}{2}$. So merely being a Dirichlet series with functional equation is not enough.

[These functions $L(s)$ that I am referring to are the $L$-functions associated to modular forms of half-integral weight, $L(s,f)$. Note that these do not have an Euler product, so there is something a bit different going on.]

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    I'd say the most obvious is that if the RH is true, then the non-trivial zeros are given by the peaks of the Fourier transform of [$\displaystyle\frac{\psi(e^u)-e^u-\ln 2\pi - \frac{1}{2} \ln(1-e^{-2u})}{e^{u/2}} e^{-a^2 u^2}$](https://en.wikipedia.org/wiki/Explicit_formulae_(L-function)#Riemann.27s_explicit_formula) as $a \to 0$, and by the [sign changes of $Z(t)$](https://en.wikipedia.org/wiki/Z_function). If the RH is false, then it is different ... – 2017-02-06
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    Also I discussed with the OP of the mysterious function $\frac{F'(s)}{F(s)} = \sum_{\rho,Re(\rho) = 1/2} \frac{1}{s-\rho}$ where $\rho$ are the zeros of some Dirichlet series with functional equation. I have the intuition this function encodes something special. – 2017-02-06