If $M$ is a pure motive over $\mathbb{Q}$, one cas define its $L$-function $L(M,s)$ which conjecturaly is a meromorphic function over $\mathbb{C}$ with finitely many poles. For example, when $M=\mathbb{Q}$ is the trivial motive, $L(\mathbb{Q},s)$ is the Riemann Zeta function $\zeta(s)$.

There is a famous concjecture saying that all values of $L(M,s)$ at integers $n$ (which are not zero or poles, say; otherwise replace value by principal value) are periods -- that is to say have real and imaginary parts that can be expressed as (according to Wikipedia's definition) differences of volumes of region in of Euclideans pace given by polynomial inequalities with rational coefficients.

This conjecture is due (if I am not mistaken) to Deligne in the case where $n$ is a so-called critical value of $L(M,s)$ and to Beilinson in general. I won't recall the definition of critical here, but I can say that for $\zeta(s)$ the critical values are $n=2,4,6,8,\dots$ and $n=-1,-3,-5,\dots$. Of course, the conjecture of Deligne in this case was known to Euler who proved that $\zeta(2n)$ is a rational times the volume of the unit ball in $\mathbb{R}^{4n}$, and $\zeta(1-2n)$ is rational, for $n \geq 1$ (the later made completely rigourous by Riemann).

Now my question:

In which case (if any) where $n$ is not a critical value is it known that L(M,n) is a period?

Here is a second question, related to the first:

Do you know a good survey on the progresses oon Beilinson's conjecture?