Let the function $f(t) = \cos(at)$, where ($0 < a < 1$). Let us define $$\zeta(z, f) = \frac{1}{\Gamma(z)} \int_0^{+\infty} \frac{t^{z-1}\cos(at)}{e^t-1}\, dt. $$ Is there a general formula that gives the values of $\zeta(2n, f)$ in terms of $n \in \mathbb{N}^*$, $a$, and the Bernoulli numbers?
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$\begingroup$ What have you tried? $\endgroup$– TravorLZHCommented Jun 16, 2023 at 9:18
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$\begingroup$ I just want to know if there is a known formula for this integral. I think now that there is no. Thanks. $\endgroup$– L.LCommented Jun 16, 2023 at 9:57
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