Dirichlet eta function
Both Dirichlet eta function and Riemann zeta function are special cases of Polylogarithm.
From this, one immediately has the functional equation of the zeta function also, as well as another means to extend the definition of eta to the entire complex plane.
With some simple algebra performed on finite sums, we can write for any complex s
The next formula, due to Lindelöf (1905), is valid over the whole complex plane, when the principal value is taken for the logarithm implicit in the exponential.