Special values of non-cm $L$-functions 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!
 A: For higher weight newforms, it seems that the conjecture you might want to look into is the Bloch-Kato conjecture.  Here are a few links that might help you get started.  The first is a survey article from 2003 (although a lot of progress has been made since then, this could be a good starting point):
https://jtnb.centre-mersenne.org/item/?id=JTNB_2003__15_1_179_0
Here is a fun paper on the distribution of zeros of certain polynomials (period polynomials) whose coefficients are built using the critical values you wish to study:
https://www.pnas.org/doi/10.1073/pnas.1600569113
Here is a high-level overview that requires a bit of background in algebraic number theory, representation theory of finite groups, and group cohomology:
https://www.claymath.org/sites/default/files/bellaiche.pdf
Here is a related Wikipedia article:
https://en.wikipedia.org/wiki/Special_values_of_L-functions
A: 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
$$(r_1,r_3,r_5,r_7,r_9)=(48,-25,20,-25,48)\omega^-$$
$$(r_0,r_2,r_4,r_6,r_8,r_{10})=(22680/691,-14,9,-9,14,-22680/691)\omega^+i$$
$$\omega^-\omega^+=(4096/691)<\Delta,\Delta>$$
