| L(s) = 1 | − 2.75·2-s + 5.57·4-s + 5-s − 2.73·7-s − 9.83·8-s − 2.75·10-s + 2.77·13-s + 7.51·14-s + 15.9·16-s + 3.78·17-s + 4.25·19-s + 5.57·20-s − 1.37·23-s + 25-s − 7.63·26-s − 15.2·28-s − 8.86·29-s + 5.46·31-s − 24.1·32-s − 10.4·34-s − 2.73·35-s − 2.30·37-s − 11.7·38-s − 9.83·40-s − 8.61·41-s − 12.3·43-s + 3.78·46-s + ⋯ |
| L(s) = 1 | − 1.94·2-s + 2.78·4-s + 0.447·5-s − 1.03·7-s − 3.47·8-s − 0.870·10-s + 0.769·13-s + 2.00·14-s + 3.98·16-s + 0.917·17-s + 0.975·19-s + 1.24·20-s − 0.287·23-s + 0.200·25-s − 1.49·26-s − 2.87·28-s − 1.64·29-s + 0.981·31-s − 4.27·32-s − 1.78·34-s − 0.461·35-s − 0.378·37-s − 1.89·38-s − 1.55·40-s − 1.34·41-s − 1.89·43-s + 0.558·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.75T + 2T^{2} \) |
| 7 | \( 1 + 2.73T + 7T^{2} \) |
| 13 | \( 1 - 2.77T + 13T^{2} \) |
| 17 | \( 1 - 3.78T + 17T^{2} \) |
| 19 | \( 1 - 4.25T + 19T^{2} \) |
| 23 | \( 1 + 1.37T + 23T^{2} \) |
| 29 | \( 1 + 8.86T + 29T^{2} \) |
| 31 | \( 1 - 5.46T + 31T^{2} \) |
| 37 | \( 1 + 2.30T + 37T^{2} \) |
| 41 | \( 1 + 8.61T + 41T^{2} \) |
| 43 | \( 1 + 12.3T + 43T^{2} \) |
| 47 | \( 1 + 5.59T + 47T^{2} \) |
| 53 | \( 1 + 0.543T + 53T^{2} \) |
| 59 | \( 1 - 0.389T + 59T^{2} \) |
| 61 | \( 1 - 14.3T + 61T^{2} \) |
| 67 | \( 1 - 3.65T + 67T^{2} \) |
| 71 | \( 1 + 9.16T + 71T^{2} \) |
| 73 | \( 1 + 8.16T + 73T^{2} \) |
| 79 | \( 1 - 3.44T + 79T^{2} \) |
| 83 | \( 1 - 4.65T + 83T^{2} \) |
| 89 | \( 1 - 6.62T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
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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
−8.106333532293413909323140175930, −7.13017375468220447468138933902, −6.71324022639154903095618920097, −5.97007560439171089169752846180, −5.34967246815699937845995411833, −3.49393407967379481907829488582, −3.11796742844885354663612765858, −1.94506499917752255694545523059, −1.17231364599469402066425997924, 0,
1.17231364599469402066425997924, 1.94506499917752255694545523059, 3.11796742844885354663612765858, 3.49393407967379481907829488582, 5.34967246815699937845995411833, 5.97007560439171089169752846180, 6.71324022639154903095618920097, 7.13017375468220447468138933902, 8.106333532293413909323140175930
