Jtdylktuy

Although $eta(1)$ is known to be $ln(2)$, I have not seen an analytically calculated value for $eta(frac{1}{2});$

$$etaleft(frac{1}{2}right) = sum_{n=1}^{infty}frac{(-1)^{(n+1)}}{sqrt{n}}$$

A web calculator gives the value to be 0.6, which seems to be right.

Isn't just
$$etaleft(frac{1}{2}right)=sum_{n=1}^inftyfrac{(-1)^{(n+1)}}{sqrt{n}}=left(1-sqrt{2}right) zeta left(frac{1}{2}right)approx 0.6048986434$$

Edit

Remember the general relation
$$etaleft(sright)=left(1-2^{1-s}right) zeta (s)$$ If you want a quick and dirty shortcut evaluation, for $0 leq s leq 1$, you could use
$$etaleft(sright)=frac 12+left( log (2)-frac{1}{2}right), s^{0.895}$$

A careful computation shows that the numerical value is
$$0.6048986434216303702472...$$
which is not $0.6$. One should be aware that the above series converge really slowly.

As Claude Leibovici indicates, one can relate its value to the Riemann's Zeta function value at $1/2$. However, as far as I know, there is no analytic formula of $zeta(1/2)$, so this is why you haven't seen an "analytically calculated value for $eta(1/2)$".

EDIT2: As pointed again in the comments by leftaroundabout, I missread the OEIS link given in the answer of R. J. Mathar. What equals $$gamma/2 + pi/4 - (1/2 + sqrt{2})log(2) + log(pi)/2,$$where $gamma$ is the Euler-Mascheroni constant, is $eta'(1/2)/eta(1/2)$ not $eta(1/2)$.

The numerical value of 0.604898... is provided in http://oeis.org/A113024 .