| L(s) = 1 | + 2-s − 2.40·3-s + 4-s − 4.08·5-s − 2.40·6-s + 0.438·7-s + 8-s + 2.77·9-s − 4.08·10-s − 4.74·11-s − 2.40·12-s − 1.49·13-s + 0.438·14-s + 9.82·15-s + 16-s − 0.270·17-s + 2.77·18-s − 1.32·19-s − 4.08·20-s − 1.05·21-s − 4.74·22-s + 7.39·23-s − 2.40·24-s + 11.7·25-s − 1.49·26-s + 0.548·27-s + 0.438·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.38·3-s + 0.5·4-s − 1.82·5-s − 0.980·6-s + 0.165·7-s + 0.353·8-s + 0.923·9-s − 1.29·10-s − 1.43·11-s − 0.693·12-s − 0.413·13-s + 0.117·14-s + 2.53·15-s + 0.250·16-s − 0.0656·17-s + 0.653·18-s − 0.303·19-s − 0.914·20-s − 0.229·21-s − 1.01·22-s + 1.54·23-s − 0.490·24-s + 2.34·25-s − 0.292·26-s + 0.105·27-s + 0.0828·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 + T \) |
| good | 3 | \( 1 + 2.40T + 3T^{2} \) |
| 5 | \( 1 + 4.08T + 5T^{2} \) |
| 7 | \( 1 - 0.438T + 7T^{2} \) |
| 11 | \( 1 + 4.74T + 11T^{2} \) |
| 13 | \( 1 + 1.49T + 13T^{2} \) |
| 17 | \( 1 + 0.270T + 17T^{2} \) |
| 19 | \( 1 + 1.32T + 19T^{2} \) |
| 23 | \( 1 - 7.39T + 23T^{2} \) |
| 29 | \( 1 - 6.66T + 29T^{2} \) |
| 37 | \( 1 + 1.40T + 37T^{2} \) |
| 41 | \( 1 - 3.81T + 41T^{2} \) |
| 43 | \( 1 - 0.726T + 43T^{2} \) |
| 47 | \( 1 - 1.04T + 47T^{2} \) |
| 53 | \( 1 - 6.49T + 53T^{2} \) |
| 59 | \( 1 + 1.18T + 59T^{2} \) |
| 61 | \( 1 + 7.18T + 61T^{2} \) |
| 67 | \( 1 + 6.27T + 67T^{2} \) |
| 71 | \( 1 - 3.32T + 71T^{2} \) |
| 73 | \( 1 + 4.65T + 73T^{2} \) |
| 79 | \( 1 - 0.885T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 14.0T + 89T^{2} \) |
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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
−7.49513632229786508782463110925, −7.02039769278099161358545032443, −6.24954207907666437884927267864, −5.34051436447613337491199077336, −4.75075684229632105195298027885, −4.46734572055289275855187313214, −3.31806363493890884638719877025, −2.65508095086239785012015388362, −0.933817694901210580429208296281, 0,
0.933817694901210580429208296281, 2.65508095086239785012015388362, 3.31806363493890884638719877025, 4.46734572055289275855187313214, 4.75075684229632105195298027885, 5.34051436447613337491199077336, 6.24954207907666437884927267864, 7.02039769278099161358545032443, 7.49513632229786508782463110925
