| L(s) = 1 | + 7.15·3-s − 9.75·5-s − 7·7-s + 24.2·9-s + 11·11-s + 1.08·13-s − 69.8·15-s − 95.3·17-s − 77.1·19-s − 50.1·21-s + 114.·23-s − 29.8·25-s − 19.5·27-s + 280.·29-s − 233.·31-s + 78.7·33-s + 68.2·35-s − 294.·37-s + 7.76·39-s − 136.·41-s − 312.·43-s − 236.·45-s − 476.·47-s + 49·49-s − 682.·51-s + 283.·53-s − 107.·55-s + ⋯ |
| L(s) = 1 | + 1.37·3-s − 0.872·5-s − 0.377·7-s + 0.898·9-s + 0.301·11-s + 0.0231·13-s − 1.20·15-s − 1.36·17-s − 0.931·19-s − 0.520·21-s + 1.03·23-s − 0.238·25-s − 0.139·27-s + 1.79·29-s − 1.35·31-s + 0.415·33-s + 0.329·35-s − 1.31·37-s + 0.0318·39-s − 0.520·41-s − 1.10·43-s − 0.784·45-s − 1.47·47-s + 0.142·49-s − 1.87·51-s + 0.733·53-s − 0.263·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 - 11T \) |
| good | 3 | \( 1 - 7.15T + 27T^{2} \) |
| 5 | \( 1 + 9.75T + 125T^{2} \) |
| 13 | \( 1 - 1.08T + 2.19e3T^{2} \) |
| 17 | \( 1 + 95.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 77.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 114.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 280.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 233.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 294.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 136.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 312.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 476.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 283.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 706.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 25.3T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.06e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 869.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 197.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 821.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 668.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 542.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.70e3T + 9.12e5T^{2} \) |
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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
−9.546001743932536888070791681230, −8.663904814661975379525515397315, −8.354039109763948754866146141171, −7.19902732407976313883589631369, −6.51826133733614055246560577275, −4.81850391319363622175355855493, −3.83994431796268828603972400349, −3.06594020999220997094359959928, −1.89862928191657351545986347557, 0,
1.89862928191657351545986347557, 3.06594020999220997094359959928, 3.83994431796268828603972400349, 4.81850391319363622175355855493, 6.51826133733614055246560577275, 7.19902732407976313883589631369, 8.354039109763948754866146141171, 8.663904814661975379525515397315, 9.546001743932536888070791681230
