| L(s) = 1 | − 2.34·2-s + 2.34·3-s + 3.47·4-s − 2.42·5-s − 5.48·6-s + 1.51·7-s − 3.46·8-s + 2.48·9-s + 5.67·10-s + 2.65·11-s + 8.14·12-s + 13-s − 3.55·14-s − 5.68·15-s + 1.14·16-s + 3.00·17-s − 5.81·18-s + 2.11·19-s − 8.44·20-s + 3.55·21-s − 6.22·22-s − 7.19·23-s − 8.11·24-s + 0.885·25-s − 2.34·26-s − 1.20·27-s + 5.28·28-s + ⋯ |
| L(s) = 1 | − 1.65·2-s + 1.35·3-s + 1.73·4-s − 1.08·5-s − 2.23·6-s + 0.573·7-s − 1.22·8-s + 0.828·9-s + 1.79·10-s + 0.801·11-s + 2.35·12-s + 0.277·13-s − 0.949·14-s − 1.46·15-s + 0.287·16-s + 0.729·17-s − 1.37·18-s + 0.485·19-s − 1.88·20-s + 0.775·21-s − 1.32·22-s − 1.49·23-s − 1.65·24-s + 0.177·25-s − 0.459·26-s − 0.232·27-s + 0.998·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 617 | \( 1 - T \) |
| good | 2 | \( 1 + 2.34T + 2T^{2} \) |
| 3 | \( 1 - 2.34T + 3T^{2} \) |
| 5 | \( 1 + 2.42T + 5T^{2} \) |
| 7 | \( 1 - 1.51T + 7T^{2} \) |
| 11 | \( 1 - 2.65T + 11T^{2} \) |
| 17 | \( 1 - 3.00T + 17T^{2} \) |
| 19 | \( 1 - 2.11T + 19T^{2} \) |
| 23 | \( 1 + 7.19T + 23T^{2} \) |
| 29 | \( 1 - 3.00T + 29T^{2} \) |
| 31 | \( 1 + 2.77T + 31T^{2} \) |
| 37 | \( 1 + 4.23T + 37T^{2} \) |
| 41 | \( 1 + 7.07T + 41T^{2} \) |
| 43 | \( 1 + 9.17T + 43T^{2} \) |
| 47 | \( 1 - 8.86T + 47T^{2} \) |
| 53 | \( 1 - 1.43T + 53T^{2} \) |
| 59 | \( 1 - 8.99T + 59T^{2} \) |
| 61 | \( 1 + 0.583T + 61T^{2} \) |
| 67 | \( 1 - 0.703T + 67T^{2} \) |
| 71 | \( 1 + 7.84T + 71T^{2} \) |
| 73 | \( 1 + 6.82T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 + 4.23T + 83T^{2} \) |
| 89 | \( 1 - 9.26T + 89T^{2} \) |
| 97 | \( 1 + 9.77T + 97T^{2} \) |
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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
−7.81417949644293030864831274501, −7.29756684432631742385982256745, −6.62087990465616473430500441313, −5.52373070119122616765807995510, −4.28355259708153227434066437704, −3.69748624421067517207541314454, −2.95517013362856840538377849072, −1.90784949568736731784651975323, −1.31842547443778555071031062267, 0,
1.31842547443778555071031062267, 1.90784949568736731784651975323, 2.95517013362856840538377849072, 3.69748624421067517207541314454, 4.28355259708153227434066437704, 5.52373070119122616765807995510, 6.62087990465616473430500441313, 7.29756684432631742385982256745, 7.81417949644293030864831274501
