| L(s) = 1 | + 5-s − 3.90·7-s + 6.09·11-s − 2.02·13-s − 2.46·17-s − 3.33·19-s − 0.830·23-s + 25-s + 2.63·29-s − 1.83·31-s − 3.90·35-s + 5.58·37-s + 6.65·41-s − 11.0·43-s − 0.166·47-s + 8.21·49-s + 12.0·53-s + 6.09·55-s + 1.73·59-s − 13.3·61-s − 2.02·65-s − 13.2·67-s − 10.8·71-s + 1.87·73-s − 23.7·77-s − 7.46·79-s − 6.80·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.47·7-s + 1.83·11-s − 0.561·13-s − 0.597·17-s − 0.765·19-s − 0.173·23-s + 0.200·25-s + 0.488·29-s − 0.328·31-s − 0.659·35-s + 0.918·37-s + 1.03·41-s − 1.68·43-s − 0.0243·47-s + 1.17·49-s + 1.65·53-s + 0.821·55-s + 0.225·59-s − 1.70·61-s − 0.251·65-s − 1.62·67-s − 1.29·71-s + 0.219·73-s − 2.70·77-s − 0.839·79-s − 0.746·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + 3.90T + 7T^{2} \) |
| 11 | \( 1 - 6.09T + 11T^{2} \) |
| 13 | \( 1 + 2.02T + 13T^{2} \) |
| 17 | \( 1 + 2.46T + 17T^{2} \) |
| 19 | \( 1 + 3.33T + 19T^{2} \) |
| 23 | \( 1 + 0.830T + 23T^{2} \) |
| 29 | \( 1 - 2.63T + 29T^{2} \) |
| 31 | \( 1 + 1.83T + 31T^{2} \) |
| 37 | \( 1 - 5.58T + 37T^{2} \) |
| 41 | \( 1 - 6.65T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 + 0.166T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 - 1.73T + 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 - 1.87T + 73T^{2} \) |
| 79 | \( 1 + 7.46T + 79T^{2} \) |
| 83 | \( 1 + 6.80T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 + 0.663T + 97T^{2} \) |
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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
−7.44354756008742751819750131984, −6.79116448239312041833423031012, −6.29649392244426488874806333433, −5.85242828563078944010608264534, −4.60417555476207775482409689858, −4.02514543726491529826392282211, −3.18717706319921795122653278243, −2.36386119914992939268171827656, −1.30724537858225723440153223838, 0,
1.30724537858225723440153223838, 2.36386119914992939268171827656, 3.18717706319921795122653278243, 4.02514543726491529826392282211, 4.60417555476207775482409689858, 5.85242828563078944010608264534, 6.29649392244426488874806333433, 6.79116448239312041833423031012, 7.44354756008742751819750131984
