| L(s) = 1 | − 3.46·5-s − 7-s + 5.46·11-s − 2·13-s − 7.46·17-s + 6.92·19-s − 5.46·23-s + 6.99·25-s + 8.92·29-s + 2.92·31-s + 3.46·35-s + 2·37-s − 4.53·41-s + 8·43-s + 2.92·47-s + 49-s − 2·53-s − 18.9·55-s − 14.9·59-s − 4.92·61-s + 6.92·65-s − 10.9·67-s − 2.53·71-s − 0.928·73-s − 5.46·77-s + 2.92·79-s − 4·83-s + ⋯ |
| L(s) = 1 | − 1.54·5-s − 0.377·7-s + 1.64·11-s − 0.554·13-s − 1.81·17-s + 1.58·19-s − 1.13·23-s + 1.39·25-s + 1.65·29-s + 0.525·31-s + 0.585·35-s + 0.328·37-s − 0.708·41-s + 1.21·43-s + 0.427·47-s + 0.142·49-s − 0.274·53-s − 2.55·55-s − 1.94·59-s − 0.630·61-s + 0.859·65-s − 1.33·67-s − 0.300·71-s − 0.108·73-s − 0.622·77-s + 0.329·79-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 3.46T + 5T^{2} \) |
| 11 | \( 1 - 5.46T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 7.46T + 17T^{2} \) |
| 19 | \( 1 - 6.92T + 19T^{2} \) |
| 23 | \( 1 + 5.46T + 23T^{2} \) |
| 29 | \( 1 - 8.92T + 29T^{2} \) |
| 31 | \( 1 - 2.92T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 4.53T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 2.92T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 14.9T + 59T^{2} \) |
| 61 | \( 1 + 4.92T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 + 2.53T + 71T^{2} \) |
| 73 | \( 1 + 0.928T + 73T^{2} \) |
| 79 | \( 1 - 2.92T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 3.46T + 89T^{2} \) |
| 97 | \( 1 - 4.92T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.992846356885766983001991000556, −7.37136082751707015251643770465, −6.67982625313640665743777995138, −6.09965788745713556658947745090, −4.67967400312300377332191244689, −4.30895469453670692609610459919, −3.53038146336328772307829236318, −2.66783289303193059459661940737, −1.21568558898854458089765081722, 0,
1.21568558898854458089765081722, 2.66783289303193059459661940737, 3.53038146336328772307829236318, 4.30895469453670692609610459919, 4.67967400312300377332191244689, 6.09965788745713556658947745090, 6.67982625313640665743777995138, 7.37136082751707015251643770465, 7.992846356885766983001991000556
