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Let $F = \mathbb{Q}(\sqrt{-d})$ with class number $h_F = 1$, and $\Gamma = \mathrm{PSL}_2(\mathfrak{O}_F)$. Let $f$ be a Maass cusp form in the $L^2$-cuspidal spectrum of the Laplace operator $\varDelta$ on $\Gamma \backslash \mathbb{H}^3$. For $\mathrm{Re} (s) > 1$ the symmetric square $L$-function attached to $ f $ is defined by \begin{equation} L (s, \mathrm{Sym}^2 f ) = \zeta_F (2s) \sum_{\mathfrak{n} \subset \mathfrak{O} } \frac {\lambda_f( \mathfrak{n}^2) } {\mathrm{N} (\mathfrak{n})^{ s} }. \end{equation}

My question is: What does the approximate functional equation (or even just the functional equation) for $L(s, \text{Sym}^2 f)$ look like? I see some results for the case when $F = \mathbb{Q}(i)$, such as in Prime Geodesic Theorem in the 3-dimensional Hyperbolic Space, but I'm wondering if there's a slightly more general version available anywhere?

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    $\begingroup$ For any L-function, can't you deduce the Gamma-factors of the sym^2 from the Gamma factors of the original L-function? $\endgroup$
    – David Farmer
    Commented Apr 25 at 18:30
  • $\begingroup$ @DavidFarmer Thanks for comment! I found the definition of the gamma factor for the Lankin-Selberg L-function in [IK,p132], so perhaps I can obtain its functional equation. But how can I derive the approximate functional equation for it? $\endgroup$
    – Misaka 16559
    Commented Apr 26 at 5:11
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    $\begingroup$ Rubinstein gives a general smoothed approximate functional equation which only needs the data in the functional equation: arxiv.org/abs/math/0412181 If you want something in the style of Hardy-Littlewood, the error term will be large so you probably can't use it to prove anything. $\endgroup$
    – David Farmer
    Commented Apr 26 at 23:20

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