How can I find information about the Dirichlet eta function in the LMFDB?

Question

I am looking for information about the Dirichlet eta function:

$$\eta(s)=\zeta(s)(1-1/2^{(s-1)})$$

How can I lookup information about the Dirichlet eta function in the LMFDB? What search words should I use? Which page should I start with?

In particular I am interested in finding the symmetric functional equation for the Dirichlet eta function.

I tried looking for the Dirichlet eta function at http://www.lmfdb.org/knowledge/ but it seems to be about the Dedekind eta function which I am not interested in.

Answer 1

I am going to answer it for you without using database since your concerns about $\eta(s)$ are rather narrow and most information can be inferred from $\zeta(s)$.

We have the functional equation for $\zeta(s)$;

$\Gamma(\frac{s}{2})\zeta(s) \pi^{-\frac{s}{2}}=\Gamma(\frac{1-s}{2})\zeta(1-s)\pi^{-\frac{1-s}{2}}=\Lambda(s)$

So if you tag along any analytic function $f(s)$ along with $\zeta(s)$, if $f(s) \neq f(1-s)$, then to preserve symmetry under the map $s \mapsto 1-s$ in the easiest way would be to tag both $f(s)$ and $f(1-s)$. In your case for $\eta(s)$, you have $f(s)=(1-\frac{2}{2^s})$ and your functional equation would be

$(1-2^s)(1-2^{1-s})\Lambda(s)=(1-2^{1-s})(1-2^{s})\Lambda(1-s)$

From there you can recover your functional equation for $\eta(s)$.