| L(s) = 1 | − 2.14·2-s + 1.70·3-s + 2.62·4-s − 3.65·6-s + 3.86·7-s − 1.33·8-s − 0.104·9-s − 3.36·11-s + 4.46·12-s + 3.60·13-s − 8.29·14-s − 2.36·16-s − 4.80·17-s + 0.224·18-s + 0.226·19-s + 6.56·21-s + 7.23·22-s − 1.91·23-s − 2.27·24-s − 7.74·26-s − 5.28·27-s + 10.1·28-s − 4.83·29-s + 3.43·31-s + 7.76·32-s − 5.72·33-s + 10.3·34-s + ⋯ |
| L(s) = 1 | − 1.52·2-s + 0.982·3-s + 1.31·4-s − 1.49·6-s + 1.45·7-s − 0.472·8-s − 0.0348·9-s − 1.01·11-s + 1.28·12-s + 0.999·13-s − 2.21·14-s − 0.592·16-s − 1.16·17-s + 0.0529·18-s + 0.0520·19-s + 1.43·21-s + 1.54·22-s − 0.400·23-s − 0.464·24-s − 1.51·26-s − 1.01·27-s + 1.91·28-s − 0.897·29-s + 0.617·31-s + 1.37·32-s − 0.997·33-s + 1.77·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 2.14T + 2T^{2} \) |
| 3 | \( 1 - 1.70T + 3T^{2} \) |
| 7 | \( 1 - 3.86T + 7T^{2} \) |
| 11 | \( 1 + 3.36T + 11T^{2} \) |
| 13 | \( 1 - 3.60T + 13T^{2} \) |
| 17 | \( 1 + 4.80T + 17T^{2} \) |
| 19 | \( 1 - 0.226T + 19T^{2} \) |
| 23 | \( 1 + 1.91T + 23T^{2} \) |
| 29 | \( 1 + 4.83T + 29T^{2} \) |
| 31 | \( 1 - 3.43T + 31T^{2} \) |
| 37 | \( 1 + 1.65T + 37T^{2} \) |
| 41 | \( 1 + 5.01T + 41T^{2} \) |
| 43 | \( 1 - 1.20T + 43T^{2} \) |
| 47 | \( 1 + 1.14T + 47T^{2} \) |
| 53 | \( 1 + 4.03T + 53T^{2} \) |
| 59 | \( 1 - 9.59T + 59T^{2} \) |
| 61 | \( 1 + 14.7T + 61T^{2} \) |
| 67 | \( 1 - 2.12T + 67T^{2} \) |
| 71 | \( 1 + 2.50T + 71T^{2} \) |
| 73 | \( 1 + 5.03T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 5.82T + 83T^{2} \) |
| 89 | \( 1 + 1.13T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−8.082904444052458715501476119051, −7.51705908639914472716308610463, −6.67179611346137238404510497528, −5.66385769403274390000736279469, −4.78509681259477238183057889209, −3.94616122412621370972297836379, −2.78928237142574923502743598512, −2.05364977362585206376288799006, −1.42940945762675649098245450475, 0,
1.42940945762675649098245450475, 2.05364977362585206376288799006, 2.78928237142574923502743598512, 3.94616122412621370972297836379, 4.78509681259477238183057889209, 5.66385769403274390000736279469, 6.67179611346137238404510497528, 7.51705908639914472716308610463, 8.082904444052458715501476119051
