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Q: What is a well-known formula of the generalized Hardy Z-function??

$\arg_0(z)=\frac{\log(z)-\log(\overline{z})}{2i}$

$a_{k,j}=\frac{1-\chi_{k,j}(-1)}{2}$

$\vartheta_{q,r}(z)=\frac{\log\Gamma(\frac14+i\frac{z}2+\frac12a_{q,r})-\log\Gamma(\frac14-i\frac{z}2+\frac12a_{q,r})}{2i}-\frac12z\log(\frac{\pi}{q})-\arg_0(L_{q,r}(\frac12))$

$Z_{q,r}(z)=e^{i\,\vartheta_{q,r}(z)}L_{q,r}(\frac12+iz)$

I could not find any formulas for the generalized Hardy Z-funcion yet! The above formula of the Riemann-Siegel theta function is a well-known formula?

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Formulas for the analogous of the Z(t) function in the case of Dirichlet L-functions are known

For example see: D. Davies, C. B Haselgrove, The evaluation of Dirichlet L-functions, Proc. of the Royal Society London. Series A 264 (1961) 122-132.

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  • $\begingroup$ Thanks@juan : The next formula of (1.5) in {Davies,Haselgrove} seems to be an approximation formula for small t. I am finding an exact formula for all complex numbers... $\endgroup$
    – Sangkyu Kim
    Commented Jun 15, 2014 at 20:03

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