| L(s) = 1 | − 1.56·2-s − 3·3-s − 5.56·4-s + 4.68·6-s + 10.2·7-s + 21.1·8-s + 9·9-s − 11·11-s + 16.6·12-s + 40.8·13-s − 16·14-s + 11.4·16-s + 98.7·17-s − 14.0·18-s − 39.6·19-s − 30.7·21-s + 17.1·22-s − 61.6·23-s − 63.5·24-s − 63.8·26-s − 27·27-s − 56.9·28-s − 149.·29-s + 54.7·31-s − 187.·32-s + 33·33-s − 154.·34-s + ⋯ |
| L(s) = 1 | − 0.552·2-s − 0.577·3-s − 0.695·4-s + 0.318·6-s + 0.553·7-s + 0.935·8-s + 0.333·9-s − 0.301·11-s + 0.401·12-s + 0.872·13-s − 0.305·14-s + 0.178·16-s + 1.40·17-s − 0.184·18-s − 0.478·19-s − 0.319·21-s + 0.166·22-s − 0.559·23-s − 0.540·24-s − 0.481·26-s − 0.192·27-s − 0.384·28-s − 0.954·29-s + 0.317·31-s − 1.03·32-s + 0.174·33-s − 0.777·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.027444351\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.027444351\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 1.56T + 8T^{2} \) |
| 7 | \( 1 - 10.2T + 343T^{2} \) |
| 13 | \( 1 - 40.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 98.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 39.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 61.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 149.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 54.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 44.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 336.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 2.36T + 7.95e4T^{2} \) |
| 47 | \( 1 - 333.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 640.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 370.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 714.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 404.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 939.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 362.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 951.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 735.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 385.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 966.T + 9.12e5T^{2} \) |
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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
−9.806986566435316226187738075720, −9.068030042095891331729383022713, −8.016825592674984079214629781883, −7.65570917597865097032426039793, −6.23415309639988120884263187498, −5.41503299734325727280676594738, −4.51972008784312345191272966679, −3.53585299880254668214246467297, −1.71164066558505707530058870290, −0.66978612872596675055054712958,
0.66978612872596675055054712958, 1.71164066558505707530058870290, 3.53585299880254668214246467297, 4.51972008784312345191272966679, 5.41503299734325727280676594738, 6.23415309639988120884263187498, 7.65570917597865097032426039793, 8.016825592674984079214629781883, 9.068030042095891331729383022713, 9.806986566435316226187738075720
