But here is the worst (or "most mysterious," depending on your mood..) thing about Siegel zeros. Our best result about Siegel zeros (excluding for present discussion Zhang's work), namely Siegel's theorem, is ineffective. That is, it says "there exists some constant C > 0 such that..." but it can tell you nothing about that constant beyond that it is positive and finite (we say that the constant is "not effectively computable from the proof").*
So then, if you try to use Siegel's theorem to prove things about primes, this ineffectivity trickles down (think "fruit of the poisoned tree"). For example, standard texts on analytic number theory include a proof of the following theorem: any sufficiently large odd integer is the sum of three primes. However, the proof in most standard texts fundamentally cannot tell you what the threshold for "sufficiently large" is, because the proof uses Siegel's theorem! In this particular case, it turns out that one can avoid Siegel's theorem, and in fact the statement "Any odd integer larger than five is the sum of three primes" is now known https://en.wikipedia.org/wiki/Goldbach%27s_weak_conjecture. But it is certainly not always possible to avoid Siegel's theorem, and Zhang's result would make so many theorems which right now involve ineffectively computable constants effective.
*Why is the constant not effectively computable? Because the proof proceeds basically like this. First: assume the Generalized Riemann Hypothesis. Then the result is trivial, Siegel zeros are exceptions to GRH and don't occur if GRH is true. Next, assume GRH is false. Take a "minimal" counterexample to GRH, and use it to "repel" or "exclude" other possible counterexamples.
