| L(s) = 1 | + 0.613·2-s − 1.62·4-s − 3.33·5-s + 1.90·7-s − 2.22·8-s − 2.04·10-s − 11-s + 0.224·13-s + 1.16·14-s + 1.88·16-s − 1.41·17-s + 7.88·19-s + 5.41·20-s − 0.613·22-s − 8.96·23-s + 6.13·25-s + 0.137·26-s − 3.08·28-s − 1.63·29-s − 6.12·31-s + 5.60·32-s − 0.868·34-s − 6.34·35-s + 8.00·37-s + 4.83·38-s + 7.41·40-s − 6.57·41-s + ⋯ |
| L(s) = 1 | + 0.433·2-s − 0.811·4-s − 1.49·5-s + 0.718·7-s − 0.785·8-s − 0.647·10-s − 0.301·11-s + 0.0621·13-s + 0.311·14-s + 0.470·16-s − 0.343·17-s + 1.80·19-s + 1.21·20-s − 0.130·22-s − 1.86·23-s + 1.22·25-s + 0.0269·26-s − 0.583·28-s − 0.302·29-s − 1.10·31-s + 0.990·32-s − 0.148·34-s − 1.07·35-s + 1.31·37-s + 0.784·38-s + 1.17·40-s − 1.02·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 0.613T + 2T^{2} \) |
| 5 | \( 1 + 3.33T + 5T^{2} \) |
| 7 | \( 1 - 1.90T + 7T^{2} \) |
| 13 | \( 1 - 0.224T + 13T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 - 7.88T + 19T^{2} \) |
| 23 | \( 1 + 8.96T + 23T^{2} \) |
| 29 | \( 1 + 1.63T + 29T^{2} \) |
| 31 | \( 1 + 6.12T + 31T^{2} \) |
| 37 | \( 1 - 8.00T + 37T^{2} \) |
| 41 | \( 1 + 6.57T + 41T^{2} \) |
| 43 | \( 1 - 9.38T + 43T^{2} \) |
| 47 | \( 1 - 3.18T + 47T^{2} \) |
| 53 | \( 1 - 8.45T + 53T^{2} \) |
| 59 | \( 1 - 8.97T + 59T^{2} \) |
| 61 | \( 1 - 5.72T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 - 8.85T + 71T^{2} \) |
| 73 | \( 1 + 2.52T + 73T^{2} \) |
| 79 | \( 1 + 9.89T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 + 9.16T + 97T^{2} \) |
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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
−7.58726863940474253308083835404, −7.02181040702658558648657113521, −5.74406152140488513670753711878, −5.37487349245583744115511198285, −4.48239770361691689113866997719, −3.96836421135472081788450188276, −3.47728151720270301462624195908, −2.41114289745391441755066124611, −1.01649094771447316560278301976, 0,
1.01649094771447316560278301976, 2.41114289745391441755066124611, 3.47728151720270301462624195908, 3.96836421135472081788450188276, 4.48239770361691689113866997719, 5.37487349245583744115511198285, 5.74406152140488513670753711878, 7.02181040702658558648657113521, 7.58726863940474253308083835404
