| L(s) = 1 | + 0.894·2-s − 3-s − 1.19·4-s + 0.474·5-s − 0.894·6-s − 1.99·7-s − 2.86·8-s + 9-s + 0.425·10-s + 4.21·11-s + 1.19·12-s + 5.77·13-s − 1.78·14-s − 0.474·15-s − 0.164·16-s − 7.31·17-s + 0.894·18-s − 1.82·19-s − 0.569·20-s + 1.99·21-s + 3.77·22-s − 23-s + 2.86·24-s − 4.77·25-s + 5.16·26-s − 27-s + 2.39·28-s + ⋯ |
| L(s) = 1 | + 0.632·2-s − 0.577·3-s − 0.599·4-s + 0.212·5-s − 0.365·6-s − 0.754·7-s − 1.01·8-s + 0.333·9-s + 0.134·10-s + 1.27·11-s + 0.346·12-s + 1.60·13-s − 0.477·14-s − 0.122·15-s − 0.0410·16-s − 1.77·17-s + 0.210·18-s − 0.419·19-s − 0.127·20-s + 0.435·21-s + 0.803·22-s − 0.208·23-s + 0.584·24-s − 0.954·25-s + 1.01·26-s − 0.192·27-s + 0.452·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 0.894T + 2T^{2} \) |
| 5 | \( 1 - 0.474T + 5T^{2} \) |
| 7 | \( 1 + 1.99T + 7T^{2} \) |
| 11 | \( 1 - 4.21T + 11T^{2} \) |
| 13 | \( 1 - 5.77T + 13T^{2} \) |
| 17 | \( 1 + 7.31T + 17T^{2} \) |
| 19 | \( 1 + 1.82T + 19T^{2} \) |
| 31 | \( 1 + 0.108T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 0.314T + 41T^{2} \) |
| 43 | \( 1 + 9.14T + 43T^{2} \) |
| 47 | \( 1 - 7.26T + 47T^{2} \) |
| 53 | \( 1 + 8.53T + 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 - 0.283T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 + 0.728T + 71T^{2} \) |
| 73 | \( 1 + 1.79T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 + 8.65T + 83T^{2} \) |
| 89 | \( 1 + 1.57T + 89T^{2} \) |
| 97 | \( 1 + 3.65T + 97T^{2} \) |
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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
−9.060031035307633703586918156059, −8.111376481546817927857854495759, −6.68290339699830549590668004581, −6.25443952119035996456130031771, −5.77869183443498050714283689070, −4.37988060569450438266719307378, −4.12562231984258512439776060503, −3.09034660780090711254112221487, −1.53007056329869075457062204685, 0,
1.53007056329869075457062204685, 3.09034660780090711254112221487, 4.12562231984258512439776060503, 4.37988060569450438266719307378, 5.77869183443498050714283689070, 6.25443952119035996456130031771, 6.68290339699830549590668004581, 8.111376481546817927857854495759, 9.060031035307633703586918156059
