| L(s) = 1 | − 11.0·2-s − 27.5·3-s + 90.9·4-s + 25·5-s + 305.·6-s − 40.7·7-s − 654.·8-s + 517.·9-s − 277.·10-s + 121·11-s − 2.50e3·12-s − 996.·13-s + 452.·14-s − 689.·15-s + 4.34e3·16-s − 474.·17-s − 5.73e3·18-s − 361·19-s + 2.27e3·20-s + 1.12e3·21-s − 1.34e3·22-s + 4.70e3·23-s + 1.80e4·24-s + 625·25-s + 1.10e4·26-s − 7.55e3·27-s − 3.71e3·28-s + ⋯ |
| L(s) = 1 | − 1.96·2-s − 1.76·3-s + 2.84·4-s + 0.447·5-s + 3.46·6-s − 0.314·7-s − 3.61·8-s + 2.12·9-s − 0.876·10-s + 0.301·11-s − 5.02·12-s − 1.63·13-s + 0.616·14-s − 0.790·15-s + 4.24·16-s − 0.398·17-s − 4.17·18-s − 0.229·19-s + 1.27·20-s + 0.556·21-s − 0.591·22-s + 1.85·23-s + 6.39·24-s + 0.200·25-s + 3.20·26-s − 1.99·27-s − 0.894·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 + 11.0T + 32T^{2} \) |
| 3 | \( 1 + 27.5T + 243T^{2} \) |
| 7 | \( 1 + 40.7T + 1.68e4T^{2} \) |
| 13 | \( 1 + 996.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 474.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 4.70e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.01e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.67e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.40e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.80e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.09e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.24e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.54e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.99e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.91e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.05T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.79e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.06e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.76e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.16e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.42e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.01e5T + 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.219092162704361545105785194864, −7.85835286732586135440320568800, −6.89166772507506331912356765592, −6.70672690433417425376377025766, −5.70113717379277913159469921203, −4.81042335232044900263687733448, −2.85710524422522638950637620044, −1.72172937965027219840136812179, −0.76840955346528454916465253815, 0,
0.76840955346528454916465253815, 1.72172937965027219840136812179, 2.85710524422522638950637620044, 4.81042335232044900263687733448, 5.70113717379277913159469921203, 6.70672690433417425376377025766, 6.89166772507506331912356765592, 7.85835286732586135440320568800, 9.219092162704361545105785194864
