Generalization $\zeta_\varphi(s)=\sum_{k=0}^\infty {\exp(I\varphi*k) \over (1+k)^s} $ 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" 
 A: jax and Gerry Myerson are correct, $ \mathrm{Li}_s ( \omega) \omega^{-1} $ (which is the formal expression you are taking about, taken as a function of $ s $), for a root of unity $ \omega $, can be written as a linear sum of Dirichlet L-functions. Let $ \omega $ be an $n $-th root of unity. Hence $ f(k) = \omega^k $ can be interpreted as a function on $ \mathbb{Z} / n \mathbb{Z} $. The group of all (Dirichlet) characters on $ \mathbb{Z} / n \mathbb{Z} $, called $ \hat{G} $, actually forms an orthonormal basis for the vector space of all functions $ \mathbb{Z} / n \mathbb{Z} \to \mathbb{C} $, with the inner product $ \langle f, g \rangle = \phi(n)^{-1}\sum f(a) \bar{g}(a)  $. Hence we can write $ f(k) $ as a linear combination of elements from $ \hat{G} $:
$$ \omega^k = \sum_{\chi \in \hat{G}} a_{\chi} \chi(k). $$
We can deduce the coefficients via what are called the Gauss sums (this is essentially a finite version of an inverse Fourier transform),
$$ a_{\chi} = \frac{1}{\phi(n)} \sum_{m=0}^{n-1} \omega^m \bar{\chi}(m). $$
When applying this decomposition of $ \omega^k $ to your original zeta expansion, we get
$$ \sum_{n=1}^\infty \frac{\omega^{n-1}}{n^s} = \frac{1}{\omega \phi(n)} \sum_{\chi \in \hat{G}} \left( \sum_{m=0}^{n-1} \omega^{m} \bar{\chi}(m)\right) L(s,\chi). $$
Functional equations of L-functions with primitive Dirichlet characters, either odd or even (which are separate cases), can be found using theta functions, but I'm not sure if we know of good formulas otherwise. I wish I could be of more help in knowing how the specific values for $ s= 0 , -1, -2, - 3, \dots $ are found, but I'm not aware of the literature there. But it does appear we do know how to calculate them somehow, because Wolfram Alpha is quite able to provide the analytic values of PolyLog[n,z] for various negative integers $ n $ and roots of unity $ \omega $. At any rate, here are a couple good references for Dirichlet characters:


*

*Prime Numbers, Chapter 6: Dirichlet Characters

*Notes on Primes in Arithmetic Progression, 1: Dirichlet Characters
A: It sounds like you'd want to look into Dirichlet characters and their corresponding $L$-series.
(edit: Oops, the sequences are in a different order. Sorry.)
A: It is the polylogarithm function $\mathrm{Li_s}(z)=\sum_{n=1}^\infty \frac{z^n}{n^s}$. 
In your notation $\zeta_\varphi(s)=\mathrm{Li_s}(e^{i \varphi})e^{-i \varphi}$.
