| L(s) = 1 | − 2.33·2-s + 3.44·4-s − 5-s − 1.04·7-s − 3.38·8-s + 2.33·10-s + 5.71·13-s + 2.44·14-s + 1.00·16-s + 1.04·17-s + 4.66·19-s − 3.44·20-s − 4.89·23-s + 25-s − 13.3·26-s − 3.61·28-s − 2.57·29-s − 2·31-s + 4.43·32-s − 2.44·34-s + 1.04·35-s − 6.89·37-s − 10.8·38-s + 3.38·40-s − 6.76·41-s + 1.04·43-s + 11.4·46-s + ⋯ |
| L(s) = 1 | − 1.65·2-s + 1.72·4-s − 0.447·5-s − 0.396·7-s − 1.19·8-s + 0.738·10-s + 1.58·13-s + 0.654·14-s + 0.250·16-s + 0.254·17-s + 1.07·19-s − 0.771·20-s − 1.02·23-s + 0.200·25-s − 2.61·26-s − 0.684·28-s − 0.477·29-s − 0.359·31-s + 0.783·32-s − 0.420·34-s + 0.177·35-s − 1.13·37-s − 1.76·38-s + 0.535·40-s − 1.05·41-s + 0.160·43-s + 1.68·46-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 | \( 1 + 2.33T + 2T^{2} \) |
| 7 | \( 1 + 1.04T + 7T^{2} \) |
| 13 | \( 1 - 5.71T + 13T^{2} \) |
| 17 | \( 1 - 1.04T + 17T^{2} \) |
| 19 | \( 1 - 4.66T + 19T^{2} \) |
| 23 | \( 1 + 4.89T + 23T^{2} \) |
| 29 | \( 1 + 2.57T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 6.89T + 37T^{2} \) |
| 41 | \( 1 + 6.76T + 41T^{2} \) |
| 43 | \( 1 - 1.04T + 43T^{2} \) |
| 47 | \( 1 - 9.79T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 7.23T + 61T^{2} \) |
| 67 | \( 1 - 8.89T + 67T^{2} \) |
| 71 | \( 1 + 1.10T + 71T^{2} \) |
| 73 | \( 1 - 1.52T + 73T^{2} \) |
| 79 | \( 1 - 9.80T + 79T^{2} \) |
| 83 | \( 1 - 3.61T + 83T^{2} \) |
| 89 | \( 1 + 3.79T + 89T^{2} \) |
| 97 | \( 1 - 16.6T + 97T^{2} \) |
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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
−7.82357760864887982601025970522, −7.50264015760261618646016929355, −6.52523844280314460439435556741, −6.05918369383212241082345996136, −4.99045895173595680234393165349, −3.75649584370811358765198395695, −3.21726047307933092892602328440, −1.92345488943174495927667102536, −1.12733855695365154973184879866, 0,
1.12733855695365154973184879866, 1.92345488943174495927667102536, 3.21726047307933092892602328440, 3.75649584370811358765198395695, 4.99045895173595680234393165349, 6.05918369383212241082345996136, 6.52523844280314460439435556741, 7.50264015760261618646016929355, 7.82357760864887982601025970522
