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Let $p$ be an odd prime, $f$ a $p$-ordinary elliptic Hecke eigenform and $L_{p}(f,s)$ the corresponding cyclotomic $p$-adic $L$-function. For a dense subset $S$ of the domain for $s$, the specialisation $L_{p}(f,s)$/period is algebraic and $p$-integral, by construction. Is there an algebraic number $c$ such that $cL_{p}(f,s)$/period is actually an algebraic integer, for all $s \in S$? We can also ask a weaker version where $S$ is replaced by an infinite subset. It seems tempting to be believe the answer to be affirmative.

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