| L(s) = 1 | + 2.11·2-s + 2.47·4-s − 5-s − 2.16·7-s + 0.996·8-s − 2.11·10-s + 1.80·11-s − 1.29·13-s − 4.57·14-s − 2.83·16-s − 0.350·17-s + 3.25·19-s − 2.47·20-s + 3.81·22-s + 0.455·23-s + 25-s − 2.73·26-s − 5.34·28-s − 2.83·29-s − 6.90·31-s − 7.98·32-s − 0.741·34-s + 2.16·35-s + 2.22·37-s + 6.89·38-s − 0.996·40-s − 9.83·41-s + ⋯ |
| L(s) = 1 | + 1.49·2-s + 1.23·4-s − 0.447·5-s − 0.817·7-s + 0.352·8-s − 0.668·10-s + 0.544·11-s − 0.358·13-s − 1.22·14-s − 0.708·16-s − 0.0850·17-s + 0.747·19-s − 0.552·20-s + 0.813·22-s + 0.0950·23-s + 0.200·25-s − 0.536·26-s − 1.01·28-s − 0.526·29-s − 1.23·31-s − 1.41·32-s − 0.127·34-s + 0.365·35-s + 0.365·37-s + 1.11·38-s − 0.157·40-s − 1.53·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| good | 2 | \( 1 - 2.11T + 2T^{2} \) |
| 7 | \( 1 + 2.16T + 7T^{2} \) |
| 11 | \( 1 - 1.80T + 11T^{2} \) |
| 13 | \( 1 + 1.29T + 13T^{2} \) |
| 17 | \( 1 + 0.350T + 17T^{2} \) |
| 19 | \( 1 - 3.25T + 19T^{2} \) |
| 23 | \( 1 - 0.455T + 23T^{2} \) |
| 29 | \( 1 + 2.83T + 29T^{2} \) |
| 31 | \( 1 + 6.90T + 31T^{2} \) |
| 37 | \( 1 - 2.22T + 37T^{2} \) |
| 41 | \( 1 + 9.83T + 41T^{2} \) |
| 43 | \( 1 + 5.54T + 43T^{2} \) |
| 47 | \( 1 + 5.24T + 47T^{2} \) |
| 53 | \( 1 + 0.434T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 + 5.58T + 67T^{2} \) |
| 71 | \( 1 - 3.93T + 71T^{2} \) |
| 73 | \( 1 + 8.14T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−7.84820373049287016455396634978, −6.97765405078665759331023595499, −6.56945321033685796922325901432, −5.67304409602534981851791472855, −5.04983347953256601368977175402, −4.21399784783113052611059127806, −3.45910660362339874551400011536, −3.01143947892240569454232874929, −1.76243823307805359997889860925, 0,
1.76243823307805359997889860925, 3.01143947892240569454232874929, 3.45910660362339874551400011536, 4.21399784783113052611059127806, 5.04983347953256601368977175402, 5.67304409602534981851791472855, 6.56945321033685796922325901432, 6.97765405078665759331023595499, 7.84820373049287016455396634978
