| L(s) = 1 | − 2.30·2-s − 0.184·3-s + 3.32·4-s − 0.542·5-s + 0.425·6-s − 0.304·7-s − 3.06·8-s − 2.96·9-s + 1.25·10-s + 11-s − 0.613·12-s − 5.72·13-s + 0.703·14-s + 0.0999·15-s + 0.425·16-s + 17-s + 6.84·18-s + 1.89·19-s − 1.80·20-s + 0.0561·21-s − 2.30·22-s − 4.78·23-s + 0.565·24-s − 4.70·25-s + 13.2·26-s + 1.09·27-s − 1.01·28-s + ⋯ |
| L(s) = 1 | − 1.63·2-s − 0.106·3-s + 1.66·4-s − 0.242·5-s + 0.173·6-s − 0.115·7-s − 1.08·8-s − 0.988·9-s + 0.396·10-s + 0.301·11-s − 0.176·12-s − 1.58·13-s + 0.188·14-s + 0.0258·15-s + 0.106·16-s + 0.242·17-s + 1.61·18-s + 0.435·19-s − 0.404·20-s + 0.0122·21-s − 0.492·22-s − 0.998·23-s + 0.115·24-s − 0.941·25-s + 2.59·26-s + 0.211·27-s − 0.191·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1492169698\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1492169698\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 2.30T + 2T^{2} \) |
| 3 | \( 1 + 0.184T + 3T^{2} \) |
| 5 | \( 1 + 0.542T + 5T^{2} \) |
| 7 | \( 1 + 0.304T + 7T^{2} \) |
| 13 | \( 1 + 5.72T + 13T^{2} \) |
| 19 | \( 1 - 1.89T + 19T^{2} \) |
| 23 | \( 1 + 4.78T + 23T^{2} \) |
| 29 | \( 1 + 8.33T + 29T^{2} \) |
| 31 | \( 1 - 0.864T + 31T^{2} \) |
| 37 | \( 1 - 6.24T + 37T^{2} \) |
| 41 | \( 1 + 3.09T + 41T^{2} \) |
| 47 | \( 1 + 7.40T + 47T^{2} \) |
| 53 | \( 1 - 0.618T + 53T^{2} \) |
| 59 | \( 1 + 6.68T + 59T^{2} \) |
| 61 | \( 1 + 0.290T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 16.0T + 71T^{2} \) |
| 73 | \( 1 + 8.64T + 73T^{2} \) |
| 79 | \( 1 - 6.41T + 79T^{2} \) |
| 83 | \( 1 + 5.43T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 - 0.218T + 97T^{2} \) |
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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
−7.82600597862359601071819840766, −7.54103656756218691338275546786, −6.68618634785914250124278583666, −5.97127538462517011478087438674, −5.21226153848338529971854223448, −4.25020168646810190864686868272, −3.19218466238993789893402744240, −2.37896941926410580727110701217, −1.59943226350994472484412493505, −0.24009622287175568802183867758,
0.24009622287175568802183867758, 1.59943226350994472484412493505, 2.37896941926410580727110701217, 3.19218466238993789893402744240, 4.25020168646810190864686868272, 5.21226153848338529971854223448, 5.97127538462517011478087438674, 6.68618634785914250124278583666, 7.54103656756218691338275546786, 7.82600597862359601071819840766
