| L(s) = 1 | + 2-s − 3.09·3-s + 4-s + 2.68·5-s − 3.09·6-s + 2.69·7-s + 8-s + 6.57·9-s + 2.68·10-s − 5.80·11-s − 3.09·12-s − 1.02·13-s + 2.69·14-s − 8.31·15-s + 16-s − 5.21·17-s + 6.57·18-s + 19-s + 2.68·20-s − 8.34·21-s − 5.80·22-s + 4.91·23-s − 3.09·24-s + 2.22·25-s − 1.02·26-s − 11.0·27-s + 2.69·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.78·3-s + 0.5·4-s + 1.20·5-s − 1.26·6-s + 1.01·7-s + 0.353·8-s + 2.19·9-s + 0.850·10-s − 1.75·11-s − 0.893·12-s − 0.282·13-s + 0.721·14-s − 2.14·15-s + 0.250·16-s − 1.26·17-s + 1.54·18-s + 0.229·19-s + 0.601·20-s − 1.82·21-s − 1.23·22-s + 1.02·23-s − 0.631·24-s + 0.445·25-s − 0.200·26-s − 2.12·27-s + 0.509·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 3.09T + 3T^{2} \) |
| 5 | \( 1 - 2.68T + 5T^{2} \) |
| 7 | \( 1 - 2.69T + 7T^{2} \) |
| 11 | \( 1 + 5.80T + 11T^{2} \) |
| 13 | \( 1 + 1.02T + 13T^{2} \) |
| 17 | \( 1 + 5.21T + 17T^{2} \) |
| 23 | \( 1 - 4.91T + 23T^{2} \) |
| 29 | \( 1 - 3.23T + 29T^{2} \) |
| 31 | \( 1 - 6.99T + 31T^{2} \) |
| 37 | \( 1 + 9.62T + 37T^{2} \) |
| 41 | \( 1 + 5.63T + 41T^{2} \) |
| 43 | \( 1 + 4.61T + 43T^{2} \) |
| 47 | \( 1 + 2.14T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 - 5.68T + 59T^{2} \) |
| 61 | \( 1 + 8.48T + 61T^{2} \) |
| 67 | \( 1 + 4.70T + 67T^{2} \) |
| 71 | \( 1 - 6.37T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 2.49T + 79T^{2} \) |
| 83 | \( 1 + 6.10T + 83T^{2} \) |
| 89 | \( 1 + 2.99T + 89T^{2} \) |
| 97 | \( 1 - 4.49T + 97T^{2} \) |
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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
−7.12746335252625328147529171037, −6.55014780853296401509137397998, −5.95823663908774545708324233250, −5.12145842521884219161221879968, −5.03884936908399574943597612838, −4.52459318902343777914712911863, −3.00689700281529067623892084836, −2.08895279891090164096940795010, −1.37546159903023060944484672814, 0,
1.37546159903023060944484672814, 2.08895279891090164096940795010, 3.00689700281529067623892084836, 4.52459318902343777914712911863, 5.03884936908399574943597612838, 5.12145842521884219161221879968, 5.95823663908774545708324233250, 6.55014780853296401509137397998, 7.12746335252625328147529171037
