For the sake of simplicity, assume $f$ is a non-cm eigenform of weight $k$ on the group $\mathrm{SL}(2, \mathbb{Z})$. Are there any known results or conjectures regarding any special values of the associated $L$-function $L(f, n)$ for any integers $n$ for weight $k > 2$? If so, what can be said about the rational factor appearing said special values? In the case of elliptic curve $L$-functions, BSD predicts the factor of $\#E(\mathbb{Q}\_{\mathrm{tors}})^{2}$ appearing in the denominator for $L(E, 1)$. I'd like to know if something similar in the form of known results or conjectures exist for $L$-functions satisfying the criteria above. I do believe they are quite mysterious, and I have been rather curious about Ramanujan's $L$-function $L(\Delta, s)$ for $s = 11$ in particular. This is something I'm far from an expert in!



Special values of non-cm $L$-functions
Since the OP is interested in particular by $L(\Delta,11)$, the relevant
theorems come into the framework of Deligne's theory of special points and
special values, while Bloch-Kato would be for $s>11$. Here the theorems are
due to Shimura and especially Manin, which for $\Delta$ (and for general eigenforms similarly) states the following: set $r\_j=(-2i\pi)^{-j-1}j!L(\Delta,j+1)$.


There exist real numbers $\omega^+$ and $\omega^-$ such that
$$(r\_1,r\_3,r\_5,r\_7,r\_9)=(48,-25,20,-25,48)\omega^-$$
$$(r\_0,r\_2,r\_4,r\_6,r\_8,r\_{10})=(22680/691,-14,9,-9,14,-22680/691)\omega^+i$$
$$\omega^-\omega^+=(4096/691)<\Delta,\Delta>$$


