This is my first question on Math.SE ; if I am wrong somewhere , please correct me
I believe that there are not any mentioned variations of a Zeta function of this form :
Where, $ w $ is a constant real, imaginary, or complex number ,
$$ \sum_{n=1}^{\infty} \ \frac{1}{n^s +w} \quad or \quad \ \sum_{n=1}^{\infty} \ \frac{(-1)^{n+1}}{n^s +w} \ \ , \ w \in\mathbb{C} $$
My Question
1) Has there been any research into this variation of a zeta function ? Any books,texts,papers or webpages I can refer to to ?
2) Does anyone know , off-hand , of how the zeros of this L-series and other L-series compare or any general properties of their convergence
3) As unlikely as it may be, does there exist an analytic continuation to an L-function that this series relates to?
I understand that this may not be considered a traditional L-series at all but I ask because it seems that this variation of an L-series could give some interesting results .
Hope someone can guide me on the right path ...
Just realized that this could be construed as a variant of Weierstrass's functions, which gives lot's of other interesting relations.
Sorry next time ill just think about it before posting. Question can be closed! thanks.