| L(s) = 1 | − 2.38·2-s − 3-s + 3.68·4-s + 2.70·5-s + 2.38·6-s + 7-s − 4.01·8-s + 9-s − 6.44·10-s + 2.12·11-s − 3.68·12-s + 0.628·13-s − 2.38·14-s − 2.70·15-s + 2.21·16-s − 0.671·17-s − 2.38·18-s − 4.69·19-s + 9.96·20-s − 21-s − 5.05·22-s − 4.25·23-s + 4.01·24-s + 2.30·25-s − 1.49·26-s − 27-s + 3.68·28-s + ⋯ |
| L(s) = 1 | − 1.68·2-s − 0.577·3-s + 1.84·4-s + 1.20·5-s + 0.973·6-s + 0.377·7-s − 1.42·8-s + 0.333·9-s − 2.03·10-s + 0.639·11-s − 1.06·12-s + 0.174·13-s − 0.637·14-s − 0.697·15-s + 0.552·16-s − 0.162·17-s − 0.562·18-s − 1.07·19-s + 2.22·20-s − 0.218·21-s − 1.07·22-s − 0.886·23-s + 0.820·24-s + 0.461·25-s − 0.294·26-s − 0.192·27-s + 0.696·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 + 2.38T + 2T^{2} \) |
| 5 | \( 1 - 2.70T + 5T^{2} \) |
| 11 | \( 1 - 2.12T + 11T^{2} \) |
| 13 | \( 1 - 0.628T + 13T^{2} \) |
| 17 | \( 1 + 0.671T + 17T^{2} \) |
| 19 | \( 1 + 4.69T + 19T^{2} \) |
| 23 | \( 1 + 4.25T + 23T^{2} \) |
| 29 | \( 1 + 8.77T + 29T^{2} \) |
| 31 | \( 1 - 8.26T + 31T^{2} \) |
| 37 | \( 1 + 2.88T + 37T^{2} \) |
| 41 | \( 1 - 0.950T + 41T^{2} \) |
| 43 | \( 1 + 3.42T + 43T^{2} \) |
| 47 | \( 1 - 7.41T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 + 8.47T + 59T^{2} \) |
| 61 | \( 1 + 8.41T + 61T^{2} \) |
| 67 | \( 1 + 0.248T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 + 6.70T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 1.52T + 83T^{2} \) |
| 89 | \( 1 - 9.66T + 89T^{2} \) |
| 97 | \( 1 + 0.763T + 97T^{2} \) |
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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
−7.56261964842007281611889696623, −6.89432454717479487686792753191, −6.13106858879893098133010677762, −5.90220190900024163992050795812, −4.78191354788164922208597449242, −3.91457983601550964705601193542, −2.49097331496970902022241782776, −1.85537989214950180968167454236, −1.22391397028776124103194928769, 0,
1.22391397028776124103194928769, 1.85537989214950180968167454236, 2.49097331496970902022241782776, 3.91457983601550964705601193542, 4.78191354788164922208597449242, 5.90220190900024163992050795812, 6.13106858879893098133010677762, 6.89432454717479487686792753191, 7.56261964842007281611889696623
