Special L-values at s = 0 encode explicit units generating class fields. The rank-one abelian case fell recently (Dasgupta–Kakde); the full conjectures stand.
The Problems
Not schoolwork — the questions that resisted Erdős, Hilbert, and everyone since. The club keeps three shelves: what is still open, what AI recently settled, and what took humanity centuries.
Tagged l-functions — 4 entries. Clear
How wildly does the Riemann zeta function grow on the critical line? Lindelöf said barely at all — a consequence of RH that may be provable even without it.
An elliptic curve has infinitely many rational points precisely when its L-function vanishes at s = 1. Deep arithmetic meets deep analysis.
All non-trivial zeros of the Riemann zeta function lie exactly on the critical line Re(s) = 1/2. The single most consequential open statement in pure mathematics.