L(s) = 1 | − 2-s − 0.618·3-s + 4-s − 2.23·5-s + 0.618·6-s + 2·7-s − 8-s − 2.61·9-s + 2.23·10-s − 1.61·11-s − 0.618·12-s − 3·13-s − 2·14-s + 1.38·15-s + 16-s − 3·17-s + 2.61·18-s − 19-s − 2.23·20-s − 1.23·21-s + 1.61·22-s + 3·23-s + 0.618·24-s + 3·26-s + 3.47·27-s + 2·28-s − 0.145·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.356·3-s + 0.5·4-s − 0.999·5-s + 0.252·6-s + 0.755·7-s − 0.353·8-s − 0.872·9-s + 0.707·10-s − 0.487·11-s − 0.178·12-s − 0.832·13-s − 0.534·14-s + 0.356·15-s + 0.250·16-s − 0.727·17-s + 0.617·18-s − 0.229·19-s − 0.499·20-s − 0.269·21-s + 0.344·22-s + 0.625·23-s + 0.126·24-s + 0.588·26-s + 0.668·27-s + 0.377·28-s − 0.0270·29-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 | \( 1 + 0.618T + 3T^{2} \) |
| 5 | \( 1 + 2.23T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 1.61T + 11T^{2} \) |
| 13 | \( 1 + 3T + 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 + 0.145T + 29T^{2} \) |
| 31 | \( 1 - 0.381T + 31T^{2} \) |
| 37 | \( 1 - 5.70T + 37T^{2} \) |
| 41 | \( 1 - 8.70T + 41T^{2} \) |
| 43 | \( 1 - 6.32T + 43T^{2} \) |
| 47 | \( 1 - 13.1T + 47T^{2} \) |
| 53 | \( 1 - 8.94T + 53T^{2} \) |
| 59 | \( 1 - 5.38T + 59T^{2} \) |
| 61 | \( 1 + 7.76T + 61T^{2} \) |
| 67 | \( 1 + 1.47T + 67T^{2} \) |
| 71 | \( 1 - 1.85T + 71T^{2} \) |
| 73 | \( 1 + 2.70T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 + 1.85T + 83T^{2} \) |
| 89 | \( 1 + 3.94T + 89T^{2} \) |
| 97 | \( 1 + 4.23T + 97T^{2} \) |
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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
−7.55201090005059828870232232128, −7.12817535973137156964291299412, −6.09729070909545258929629001577, −5.48184227509237149910406286476, −4.62956624010254801722489925914, −4.03598559035604298697352667156, −2.80276284129701785869264019402, −2.31920242341318240746255553032, −0.910625815505891104699746177698, 0,
0.910625815505891104699746177698, 2.31920242341318240746255553032, 2.80276284129701785869264019402, 4.03598559035604298697352667156, 4.62956624010254801722489925914, 5.48184227509237149910406286476, 6.09729070909545258929629001577, 7.12817535973137156964291299412, 7.55201090005059828870232232128