The short version of my question goes: What is known to follow from the existence of Siegel zeros?

A longer version to give an idea of what I have in mind: The "expectional zeros" of course first cropped up in the work of Deuring and Heilbronn on the Gauss class number problem, the finitude of discriminants $D$ where $h(D) = 1$ (or $h(D) = c$ in general) is shown to follow from the failure of the RH (Deuring-Mordell) or, more generally, the failure of the GRH (Heilbronn). Since the finitude also follows (with good bounds) on the assumption of the GRH (Hecke-Landau) this settles the result unconditionally. (A more careful analysis by Heilbronn-Linfoot gives explicit bounds so that $h(D) = 1$ can happen at most 10 times.) Siegel later crystallised out the peculiar case of the exceptional real zero and proved the result in a very sharp form.

Linnik then used these ideas to show that $P(a,q) \ll q^L$, $P(a,q)$ being the least prime $\equiv a \pmod q$ and $L$ an explicit constant. Again this was known to follow on the GRH (with $L < 2 + \varepsilon$), hence the result is unconditional.

Iwaniec [Conversations on the Exceptional Character, p. 118 ] mentions on the other hand that Siegel zeros can be exploited to give even stronger results than what is available on the (presumably true) GRH; for example it follows that $L < 2$, hence improving on the bound known on the GRH. This is related to earlier work by Heath-Brown showing that the existence of exceptional zeros implies both that $L < 3$ (which is better than what is known unconditionally, although not better than the $L < 2 + \varepsilon$ that follows on the GRH) and the existence of infinitely many twin primes.

The results of Heath-Brown and Iwaniec, however, assumes an infinite number of exceptional zeros, to increasing modulus $q$, to derive results stronger than what is known unconditionally (or even on the GRH). What I would like to know is if similar results are proved in the literature working only from a single exceptional zero (or even from the weaker assumption of the mere existence of *any* zeros off the critical line, like Deuring and Heilbronn did)? Putting $\Theta =$ supremum of the real parts of the zeros, we have of course that $1/2 < \Theta$ (i.e. the failure of the RH) would mess up the approximate formulas for different prime counting functions (e.g. by a theorem of Schmidt we have $\Pi(x) - {\rm li} (x) = \Omega_\pm(x^{\Theta-\varepsilon})$, thus $= \Omega_\pm(x^{1/2})$ in case RH fails), but as long as the mere existence of *any* zero off the line is assumed (not some horrifying disaster like $\Theta = 1$), I know of no results stronger than the unconditionally known irregularites in the approximation (e.g. from Littlewood's theorem). The assumption of a Siegel zero of course is an assumption of such a special failure of the GRH on the other hand, so perhaps I'm missing some rather obvious examples?