| L(s) = 1 | + 1.73·5-s − 7-s − 1.26·11-s − 13-s − 0.464·17-s − 4.19·19-s − 4.73·23-s − 2.00·25-s − 0.464·29-s + 6.19·31-s − 1.73·35-s + 7.19·37-s + 9.46·41-s + 8.39·43-s + 8.19·47-s + 49-s + 2.53·53-s − 2.19·55-s − 2.19·59-s − 11.3·61-s − 1.73·65-s + 6.19·67-s − 16.3·71-s + 1.19·73-s + 1.26·77-s − 4.19·79-s − 4.73·83-s + ⋯ |
| L(s) = 1 | + 0.774·5-s − 0.377·7-s − 0.382·11-s − 0.277·13-s − 0.112·17-s − 0.962·19-s − 0.986·23-s − 0.400·25-s − 0.0861·29-s + 1.11·31-s − 0.292·35-s + 1.18·37-s + 1.47·41-s + 1.27·43-s + 1.19·47-s + 0.142·49-s + 0.348·53-s − 0.296·55-s − 0.285·59-s − 1.45·61-s − 0.214·65-s + 0.756·67-s − 1.94·71-s + 0.139·73-s + 0.144·77-s − 0.472·79-s − 0.519·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 1.73T + 5T^{2} \) |
| 11 | \( 1 + 1.26T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + 0.464T + 17T^{2} \) |
| 19 | \( 1 + 4.19T + 19T^{2} \) |
| 23 | \( 1 + 4.73T + 23T^{2} \) |
| 29 | \( 1 + 0.464T + 29T^{2} \) |
| 31 | \( 1 - 6.19T + 31T^{2} \) |
| 37 | \( 1 - 7.19T + 37T^{2} \) |
| 41 | \( 1 - 9.46T + 41T^{2} \) |
| 43 | \( 1 - 8.39T + 43T^{2} \) |
| 47 | \( 1 - 8.19T + 47T^{2} \) |
| 53 | \( 1 - 2.53T + 53T^{2} \) |
| 59 | \( 1 + 2.19T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 - 6.19T + 67T^{2} \) |
| 71 | \( 1 + 16.3T + 71T^{2} \) |
| 73 | \( 1 - 1.19T + 73T^{2} \) |
| 79 | \( 1 + 4.19T + 79T^{2} \) |
| 83 | \( 1 + 4.73T + 83T^{2} \) |
| 89 | \( 1 + 5.53T + 89T^{2} \) |
| 97 | \( 1 + 16T + 97T^{2} \) |
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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
−7.44483706527928665782066126390, −6.56841523929338273404171429498, −5.93265545218673359475209110602, −5.61055511231953190925961314874, −4.39836640893663230855308971034, −4.09076799834076615630392633369, −2.69580525922008977874003656555, −2.42772176024631753173523477444, −1.28179273701635474903329703532, 0,
1.28179273701635474903329703532, 2.42772176024631753173523477444, 2.69580525922008977874003656555, 4.09076799834076615630392633369, 4.39836640893663230855308971034, 5.61055511231953190925961314874, 5.93265545218673359475209110602, 6.56841523929338273404171429498, 7.44483706527928665782066126390
