This is more a reference-request for some fiddling/exploration with the $\zeta$-function. In expressing the $\zeta$ and the alternating $\zeta$ (="$\eta$") in terms of matrixoperations I asked myself, what we get, if we generalize the idea of the alternating signs to cofactors from the complex unit-circle.
$ \zeta_\varphi(s)=\sum_{k=0}^\infty {\exp(I\varphi*k) \over (1+k)^s} $
With this the usual "alternating $\zeta$" (or "$\eta$")- function were identified with $\varphi=\pi$, so each second term of the $\zeta$-series has the cofactor of $-1$. I looked at $\varphi=\pi/2$ and $\varphi=\pi/4$ so far. The $\zeta_{\pi/2}(s)$ for $s=[0,-1,-2,-3,-4,\ldots ]$ are for instance
$[1/2*(1+I), 1/2I, -1/2, -1I, 5/2, 8I, -61/2, -136I, 1385/2, \ldots]$.
But after this,... well a better approach is first to ask here for known results/discussion because involving $\zeta$ means usually, that very likely someone has looked at something like this before....(For instance, it seems we have analogues to the bernoulli-polynomials and possibly there is also an analogon to the Euler-Maclaurin-formula)
Since Gerry asked for the computation and I lost him at $M_\varphi$ I'll put some more explanation here.
I express the problem in my matrix-notation (which is admittedly a very private notation lacking rigor but hopefully gives a clue of what I'm doing). For this the main ingredient is the ubiquituous occurence of a vector V(x) (ideally of infinite size as all matrices involved) which is meant to denote the reference to the argument x in a taylor-series of a function f(x). Here the coefficients of the formal powerseries are collected in some vector A , V(x) means $\small [1,x,x^2,x^3,...x^n]$ (ideally $n\to\infty$ and f(x) is principally expressed as dot- or matrixproduct $ \small f(x) = A \cdot V(x)$. Then this notation allows to formulate the required manipulations on the formal powerseries by vector-operations, matrixproducts, matrixpowers and inversion.
Let P be the lower triangular Pascalmatrix, containing the binomial-coefficients. Then the binomial-theorem allows to write $ \small P \cdot V(x) = V(x+1) $ $ \small P^2 \cdot V(x) = V(x+2) $ and so on. Then we can do the linear combination
$ \small (P^0 + P + P^2 + ... + P^k) \cdot V(1) = V(1) + V(2) + .... + V(1+k) = S(1,k) $
Here we get in all rows of S the sums-of-like-powers of exponents 0,1,2,3,... and so on, simultaneously.
Using the alternating signed sum we can extend the sum to infinite index k getting
$ \small \begin{eqnarray} AS(1) &=& V(1)-V(2)+V(3)-...\\\ &=& (P^0-P+P^2-P^3+...-...)\cdot V(1) \\\ &=& (Id + P)^{-1} \cdot V(1) \\\ &=& H \cdot V(1) \end{eqnarray} $
which involves the closed-form-formula for geometric series for a matrix-argument.
In each row of the result AS(1) we get now the Dirichlet $\eta$ at the nonpositive integer index according to the rowindex. Also the matrix $ \small H = (Id+P)^{-1}$ contains that $\eta$'s and moreover the rows describe a modification of the bernoulli-polynomials adapted to the problem of alternating summing of like powers.
The non-alternating sum, resulting in $\zeta$-values (usually expressed in terms of bernoulli-numbers) cannot be taken this way because $ \small Id - P$ cannot be inverted (but there is a workaround such that we still get a solution).
Now the generalization (for which I ask for references) is $ \small S_\varphi (1) = V(1) + z V(2) + z^2 V(3) + z^3 V(4)+ z^4 V(5) + ... $ where $\small z = \exp(I \cdot \varphi) $ lies on the complex unit-circle. Thus $ \small AS(1) = S_{\pi}(1) $ and generally $ \small \begin{eqnarray} S_{\varphi}(1) &=& (P^0 + z \cdot P^1 + z^2 \cdot P^2 + z^3 \cdot P^3 + ...) \cdot V(1) \\\ & =& (Id - z \cdot P)^{-1} \cdot V(1) \\\ &=&M_\varphi \cdot V(1) \end{eqnarray} $ where the matrix $\small M_\varphi $ can be computed as long as $\varphi \ne 0$
The $\zeta_\varphi(k)$ for k=[0,-1,-2,-3,...] can now be taken from the according row of $\small S_\varphi(1) $. I assume that using the matrix $ \small M_\varphi$ we can construct analogues of the bernoulli-polynomials to compute the $ \small \zeta_\varphi(s)$ at real or complex s, which is what I was referring to in my above question.
[update] the idea of taking the "geometric series" for some matrix as I did it here $\small M_\varphi = (Id - z \cdot P)^{-1} $ is known as "Neumann-series"