TITLE: Are there examples that we know a number is a rational number but we do not know what is its numerator and denominator?
QUESTION [8 upvotes]: Are there examples that we know a number is a rational number but we do not know what is its numerator and denominator?
In order to say clearly, this number should given by a certain formula, such as $\sum_{i=1}^\infty f(n)$ (or $\int_0^\infty f(x)dx$)where $f(n)$ is a certain function so that we can calculate $f(n)$ for any given integer number $n$ (or real number $x$). Hence we avoid the answers like "the least even integer $N$ which makes Goldbach conjecture not true" or "the age when I get married".
Thanks in advance!

REPLY [3 votes]: Let $E$ be an elliptic curve over $\mathbb Q$ of (algebraic) rank $0$. Let $L(E,s)$ be its associated $L$-function. Then the quantity $L(E,1)/\Omega_E$, where $\Omega_E$ is the period of $E$, is known to be rational, even if we don't know its numerator and denominator.