Landau-Siegel zeros: Why can't they be found?

Question

Let me situate myself in this discussion: I'm not a mathematician or someone who had a comprehensive mathematical training. Yet I do have a great interest in Mathematics and I've been recently reading about the Generalized Riemann Hypothesis and the zeros of the zeta function $\zeta (s)$. I also learnt about the Landau-Siegel zeros which, as I understood, are real (non-complex) zeros of $\zeta(s)$ very close to $s = 1$ (to the left). The zeta function is well defined, continuous and differentiable (or isn't it?) in the west neighbourhood of $s = 1$. With the computers we have today and with the bunch of numerical methods known, finding a root should not be an unsurmountable task for an applied mathematician, or so I believe. How come no one has ever found one of these roots yet? Right, they may not exist, but their non-existence is a consequence of not finding them. I know numerical methods have their limitations and are no substitute to a formal proof, but can't they be explored somehow? Or are they being explored $-$ it's just me who am unaware of those endeavours?

Answer 1

Landau-Siegel zeros do not refer to zeros of $\zeta(s)$, but instead to (potential) zeros of Dirichlet $L$-functions associated to quadratic Dirichlet characters. Specifically, if $\chi$ is a real quadratic character mod $D$, then the question is to determine if there are zeros within approximately $1/\log(D)$ of $s = 1$ in the Dirichlet $L$-function $$\sum_{n \geq 1} \frac{\chi(n)}{n^s}.$$ People have computed an enormous number of zeros for an enormous number of quadratic Dirichlet $L$-functions. It is possible (and not even particularly challenging) to show that there are no zeros with real $s$, $\frac{1}{2} < s < 1$, for any individual Dirichlet $L$-function. But the challenge is that there are infinitely many Dirichlet $L$-functions, and we can't check them all one by one.