Is it even related to the Gamma Function?

Calculus Level 4

Let \(f(z, s)\) be an analytic function of \(z\) defined for \(\Re (z) >0\), and by the summation formula defined for \(0<x<1\) and \(s > 0\) as - \[ \large f(x,s) = \sum_{n=0}^\infty \frac{ (-1)^nx^{n+s}}{n+s}\] Using analytic continuation and extending the domain of \(x \) to all positive real numbers in \(\dfrac{\partial}{\partial x} f(x,s)\), if \( y = \displaystyle \int_0^\infty \frac{\partial}{\partial x}f\left(x,\frac16\right) dx\), find \(\cos\left(\dfrac{3y}{2}\right)\). Give your answer to two decimal places \(\)

Relevant wiki: Gamma function

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