Concerning the classical normalized Eisenstein series Earlier I asked this question. As of today, it has not been answered. Yet still, I have a follow-up question: In general, how does one express $E_4(\tau)$ and $E_6(\tau)$ in closed form for special values of $\tau$? What is the standard method? For example, how does one explicitly evaluate $E_4(\sqrt{-7})$ and $E_6(\sqrt{-7})$? I know these can be expressed in closed form (in terms of gamma functions), but is there some classical result which allows one to do it for certain $\tau$?
Any help would be greatly appreciated.
 A: These sorts of formulas are known, I think.  Particular cases go back to Hurwitz (in the late 1800's) and perhaps even further back, to Gauss (maybe), Eisenstein, and others.   The general case, though, is due (as far as I know) to Damerell.
Actually, what I am now going to discuss involves just getting the transcendental 
factors correct, rather than pinning the number down precisely.  But I'm pretty
sure that, at least in particular cases like $\sqrt{-7}$, this "pinning down" would
have been known, although perhaps not well-known.
Damerell's paper (his thesis, I think) is here.  I find it pretty
hard to read, though, since it doesn't use modern notation and terminology
for modular forms.
I learnt about Damerell's results from the papers of Katz, in which he constructs
his (so-called) two-variable $p$-adic $L$-functions for Hecke characters.  He has
a beautiful paper called "$p$-adic $L$-functions via moduli" which I couldn't find
on line.   But here is his 1978 ICM address, with Damerell's result stated on the second page: if you take
$r = 0$ in $A(k,r)$, then you have a sum of powers of elements in the ring of
integers of an imaginary quadratic field, which is exactly the value of an Eisenstein series.   (In the case where of $\sqrt{-7}$, I guess your value of $E_4$
is summing over elements of $\mathbb Z[\sqrt{-7}]$ rather than $\mathbb Z[(1 + \sqrt{-7})/2]$ so there is a slight difference, which I think should be easy to sort
out.  I also think Damerell's paper itself actually handle non-maximal orders 
directly.)
Of course you need to know the value of $\Omega$, the relevant period: but this is given by the Chowla--Selberg formula.  What you can see, then, is that Damerell's result gives a description of the Eisenstein values you are interested in terms of a 
product of $\Gamma$-values and powers of $\pi$ with an algebraic number which (at least in the form of the result stated by Katz) is not pinned down.
This formula is related to special values of certain $L$-functions, and so Damerell's result is a special case of a statement conjectured by Deligne.
Pinning down the algebraic factor in general is the topic of the Bloch--Kato
conjecture.  In this particular case, I don't know what is known about it.
Here is a paper discussing
these kind of rationality/algebraicity results, connecting the particular case of 
CM elliptic curves (or CM abelian varieties, more generally) to Deligne's general
framework.
You might also want to look at Weil's book on Elliptic functions according to Eisenstein and Kronecker, which might have some relevant classical information.

In conclusion, I would compare your method with known proofs of the Chowla--Selberg formula, of Damerell's theorem, of Lerch's formula, and of other related results, to see whether you have
rediscovered a known approach.  If not, i.e. if you've found a new approach to
this circle of ideas, that's definitely interesting.  (Even if it's not new,
it's still interesting! --- but just not new.)
A: The functions $E_{4},E_{6}$ are modular form equivalents of the functions $Q, R$ of Ramanujan given by $$Q(q) =1+240\sum_{i=1}^{\infty}\frac{i^{3}q^{i}}{1-q^{i}}\tag{1}$$ and $$R(q) =1-504\sum_{i=1}^{\infty}\frac{i^{5}q^{i}}{1-q^{i}}\tag{2}$$ The link between $q$ and $\tau$ is given by $q=\exp(2\pi i\tau) $ and because of that factor $2$ in $2\pi i\tau$ it makes sense to use the relations $$Q(q^{2})=\left(\frac{2K}{\pi}\right)^{4}(1-k^{2}+k^{4})\tag{3}$$ and $$R(q^{2})=\left(\frac{2K}{\pi}\right)^{6}(1+k^{2})(1-2k^{2})\left(1-\frac{k^{2}}{2}\right)\tag{4}$$ where $K, E$ are Complete elliptic integrals with modulus $k$ given $$k=\frac{\vartheta_{2}^{2}(q)}{\vartheta_{3}^{2}(q)}$$ The $\vartheta $'s above are theta functions of Jacobi. We now have $$E_{4}(\sqrt{-n})=Q(q^{2}),E_{6}(\sqrt{-n})=R(q^{2})\tag{5}$$ where $q=e^{-\pi\sqrt{n}}$.
Using integral transformations Jacobi proved that if $n$ is a positive rational then $k$ is an algebraic number. The simplest case is $n=1,k=1/\sqrt{2}$ and such values of $k$ are called singular moduli and a table of such values of $k$ is readily available. The desired closed form requires us to evaluate the complete elliptic integral $K$ for these values of modulus $k$. For $k=1/\sqrt{2}$ this is readily done via Gamma functions and we have $$K(1/\sqrt{2})=\frac{\Gamma^{2} (1/4)}{4\sqrt{\pi}}$$ and if $n$ is a perfect square then one can show that $K(k) /K(1/\sqrt{2})$ is algebraic. For some other values of $n$ it is possible to express $K$ in terms of Gamma function (see mathworld page for some evaluations). And then it is possible to have an explicit closed form evaluation of Eisenstein series.
