L(s) = 1 | − 2-s + 4-s + 2.72·5-s − 2.55·7-s − 8-s − 2.72·10-s + 0.632·11-s − 2.81·13-s + 2.55·14-s + 16-s − 1.96·17-s − 0.973·19-s + 2.72·20-s − 0.632·22-s + 7.05·23-s + 2.42·25-s + 2.81·26-s − 2.55·28-s − 6.39·29-s + 0.0203·31-s − 32-s + 1.96·34-s − 6.95·35-s + 5.11·37-s + 0.973·38-s − 2.72·40-s + 4.86·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.21·5-s − 0.965·7-s − 0.353·8-s − 0.861·10-s + 0.190·11-s − 0.780·13-s + 0.682·14-s + 0.250·16-s − 0.476·17-s − 0.223·19-s + 0.609·20-s − 0.134·22-s + 1.47·23-s + 0.484·25-s + 0.552·26-s − 0.482·28-s − 1.18·29-s + 0.00365·31-s − 0.176·32-s + 0.336·34-s − 1.17·35-s + 0.841·37-s + 0.157·38-s − 0.430·40-s + 0.759·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 149 | \( 1 + T \) |
good | 5 | \( 1 - 2.72T + 5T^{2} \) |
| 7 | \( 1 + 2.55T + 7T^{2} \) |
| 11 | \( 1 - 0.632T + 11T^{2} \) |
| 13 | \( 1 + 2.81T + 13T^{2} \) |
| 17 | \( 1 + 1.96T + 17T^{2} \) |
| 19 | \( 1 + 0.973T + 19T^{2} \) |
| 23 | \( 1 - 7.05T + 23T^{2} \) |
| 29 | \( 1 + 6.39T + 29T^{2} \) |
| 31 | \( 1 - 0.0203T + 31T^{2} \) |
| 37 | \( 1 - 5.11T + 37T^{2} \) |
| 41 | \( 1 - 4.86T + 41T^{2} \) |
| 43 | \( 1 + 3.52T + 43T^{2} \) |
| 47 | \( 1 + 8.40T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 7.92T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 + 1.39T + 67T^{2} \) |
| 71 | \( 1 + 2.93T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 5.40T + 79T^{2} \) |
| 83 | \( 1 + 5.25T + 83T^{2} \) |
| 89 | \( 1 + 1.53T + 89T^{2} \) |
| 97 | \( 1 + 4.95T + 97T^{2} \) |
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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
−7.35203838816013178754631602380, −6.81659595477025662916026124591, −6.24481434343310082035582084628, −5.58101255319763745757113263923, −4.83820330575539400816730846386, −3.72874366449842570951249087124, −2.78902741930866563708806568390, −2.23199959816598432611423595906, −1.24597008144378372440883050941, 0,
1.24597008144378372440883050941, 2.23199959816598432611423595906, 2.78902741930866563708806568390, 3.72874366449842570951249087124, 4.83820330575539400816730846386, 5.58101255319763745757113263923, 6.24481434343310082035582084628, 6.81659595477025662916026124591, 7.35203838816013178754631602380