<s>[INST] What is the exact value of $\eta(6i)$? Let $\eta(\tau)$ be the Dedekind eta function. In his Lost Notebook, Ramanujan played around with a related function and came up with some of the nice evaluations,
$$\begin{aligned}
\eta(i) &= \frac{1}{2} \frac{\Gamma\big(\tfrac{1}{4}\big)}{\pi^{3/4}}\\
\eta(2i) &= \frac{1}{2^{11/8}} \frac{\Gamma\big(\tfrac{1}{4}\big)}{\pi^{3/4}}\\
\eta(3i) &= \frac{1}{2\cdot 3^{3/8}} \frac{1}{(2+\sqrt{3})^{1/12}} \frac{\Gamma\big(\tfrac{1}{4}\big)}{\pi^{3/4}}\\
\eta(4i) &= \frac{1}{2^{29/16}} \frac{1}{(1+\sqrt{2})^{1/4}} \frac{\Gamma\big(\tfrac{1}{4}\big)}{\pi^{3/4}}\\
\eta(5i) &= \frac{1}{2\sqrt{5}}\left(\tfrac{1+\sqrt{5}}{2}\right)^{-1/2}\, \frac{\Gamma\big(\tfrac{1}{4}\big)}{\pi^{3/4}}\\
\eta(6i) &=\; \color{red}{??}\\
\eta(7i) &= \frac{1}{2\sqrt{7}}\left(-\tfrac{7}{2}+\sqrt{7}+\tfrac{1}{2}\sqrt{-7+4\sqrt{7}} \right)^{{1/4}}\, \frac{\Gamma\big(\tfrac{1}{4}\big)}{\pi^{3/4}}\\
\eta(8i) &= \frac{1}{2^{73/32}} \frac{(-1+\sqrt[4]{2})^{1/2}}{(1+\sqrt{2})^{1/8}} \frac{\Gamma\big(\tfrac{1}{4}\big)}{\pi^{3/4}}\\
\eta(16i) &= \frac{1}{2^{177/64}} \frac{(-1+\sqrt[4]{2})^{1/4}}{(1+\sqrt{2})^{1/16}} \left(-2^{5/8}+\sqrt{1+\sqrt{2}}\right)^{1/2}\,\frac{\Gamma\big(\tfrac{1}{4}\big)}{\pi^{3/4}}\end{aligned}$$
with the higher ones $>4$ added by this OP. (Note the powers of $2$.)

Questions: 
  
  
*
  
*Similar to the others, what is the exact value of $\eta(6i)$?
  
*Is it true that the function,
  $$F(\sqrt{-N}) = \frac{\pi^{3/4}}{\Gamma\big(\tfrac{1}{4}\big)}\,\eta(\sqrt{-N}) $$
  is an algebraic number only if $N$ is a square?

P.S. It seems strange there is a function that yields an algebraic number for square input $N$ and a transcendental number for non-square $N$. (Are there well-known functions like that?) For an example of non-square $N$, we have,
$$\eta(\sqrt{-3}) =  \frac{3^{1/8}}{2^{4/3}} \frac{\Gamma\big(\tfrac{1}{3}\big)^{3/2}}{\pi} = 0.63542\dots$$
and $F(\sqrt{-3})$ seems to be transcendental.
 [/INST] Since we know the value of $\eta(3i)$, the point is just to compute the value of the product:
$$ \prod_{n\geq 0}(1+e^{-6\pi n})=\exp\sum_{n\geq 0}\log\left(1+e^{-6\pi n}\right)=\exp\sum_{n\geq 0}\int_{n}^{n+1}\frac{6\pi n}{1+e^{6\pi s}}\,ds$$
where:
$$\sum_{n\geq 0}\int_{n}^{n+1}\frac{6\pi n}{1+e^{6\pi s}}\,ds = \int_{0}^{+\infty}\frac{6\pi s\,}{1+e^{6\pi s}}-\int_{0}^{+\infty}\frac{6\pi\{s\}}{1+e^{6\pi s}}$$
and the first integral in the RHS equals $\frac{\pi}{72}$ by the residue theorem, while expanding the fractional part as its Fourier series, $\{s\}=\frac{1}{2}-\sum_{n\geq 1}\frac{\sin(2\pi n s)}{\pi n}$, we get:
$$\begin{eqnarray*}\int_{0}^{+\infty}\frac{6\pi\{s\}\,ds}{1+e^{6\pi s}}&=&\frac{\log 2}{2}-\sum_{n\geq 1}\int_{0}^{+\infty}\frac{6 \sin(2\pi n s)}{n(e^{6\pi s}+1)}\,ds\\&=&\frac{\log 2}{2}-\sum_{n\geq 1}\frac{6}{n}\sum_{m\geq 0}(-1)^m\int_{0}^{+\infty}\sin(2\pi n s)\,e^{-6\pi m s}\,ds\\&=&\frac{\log 2}{2}-\frac{3}{\pi}\sum_{n\geq 1}\sum_{m\geq 0}\frac{(-1)^m}{9 m^2+n^2}\\&=&\frac{\log 2}{2}-\frac{\pi}{2}-\frac{3}{\pi}\sum_{m,n\geq 1}\frac{(-1)^m}{9m^2+n^2}\end{eqnarray*}$$
and the last series just depends on the number of ways to represent a positive integer $\not\equiv 2\pmod{3}$ through the binary quadratic form $n^2+9m^2$: it is, with minor manipulations, just a Dirichlet convolution. I have just applied the same techniques of this answer, just in reverse. 
This shows a clear connection between the evaluation of the Dedekind eta function at quadratic irrationals and the class number problem: $\eta(\sqrt{-N})$ depends on $\sum_{n\geq 1}(-1)^n\frac{r(n)}{n}$, where $r(n)$ counts the number of ways to represent $n$ as $a^2+Nb^2$. If $N$ is a square or $a^2+Nb^2$ is the only reduced quadratic form of discriminant $-4N$ (class number one) we may explicitly compute such series, and it turns out that $F(\sqrt{-N})$ is an algebraic number. Otherwise, $\sum_{n\geq 1}(-1)^n\frac{r(n)}{n}$ is not even a convolution of Dirichlet series, hence your conjecture is very likely to hold.
Ultimately, the computation of $\eta(6i)$ can be carried on by recalling that:
$$j(\tau)=\left(\left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^8+2^8\left(\frac{\eta(2\tau)}{\eta(\tau)}\right)^{16}\right)^{3} $$
and by computing the Klein $j$-invariant $j(3i)$. The Wikipedia page gives:
$$ j(3i) = \frac{1}{27}(2+\sqrt{3})^2(21+20\sqrt{3})^3.$$
</s>