| L(s) = 1 | + 3.95·2-s − 0.384·3-s + 7.66·4-s − 1.52·6-s − 27.0·7-s − 1.31·8-s − 26.8·9-s + 11·11-s − 2.94·12-s − 82.9·13-s − 106.·14-s − 66.5·16-s + 100.·17-s − 106.·18-s + 38.6·19-s + 10.3·21-s + 43.5·22-s − 19.5·23-s + 0.505·24-s − 328.·26-s + 20.7·27-s − 207.·28-s − 100.·29-s + 121.·31-s − 252.·32-s − 4.22·33-s + 397.·34-s + ⋯ |
| L(s) = 1 | + 1.39·2-s − 0.0739·3-s + 0.958·4-s − 0.103·6-s − 1.45·7-s − 0.0581·8-s − 0.994·9-s + 0.301·11-s − 0.0709·12-s − 1.76·13-s − 2.04·14-s − 1.03·16-s + 1.43·17-s − 1.39·18-s + 0.466·19-s + 0.107·21-s + 0.421·22-s − 0.176·23-s + 0.00430·24-s − 2.47·26-s + 0.147·27-s − 1.39·28-s − 0.641·29-s + 0.706·31-s − 1.39·32-s − 0.0223·33-s + 2.00·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 3.95T + 8T^{2} \) |
| 3 | \( 1 + 0.384T + 27T^{2} \) |
| 7 | \( 1 + 27.0T + 343T^{2} \) |
| 13 | \( 1 + 82.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 100.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 38.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 19.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 100.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 121.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 347.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 108.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 268.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 568.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 603.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 32.7T + 2.05e5T^{2} \) |
| 61 | \( 1 - 156.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 745.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 264.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 334.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.28e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 127.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 615.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 579.T + 9.12e5T^{2} \) |
| 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
−11.49008287061599717915661941208, −9.914890980295300914750545643334, −9.384154804039525497107038611684, −7.76835114972105730932131093729, −6.56554475582359428143808892013, −5.78338104169365385774168160299, −4.83023328946922227243885703316, −3.39819294510870447920749526959, −2.75261919624950895974588341350, 0,
2.75261919624950895974588341350, 3.39819294510870447920749526959, 4.83023328946922227243885703316, 5.78338104169365385774168160299, 6.56554475582359428143808892013, 7.76835114972105730932131093729, 9.384154804039525497107038611684, 9.914890980295300914750545643334, 11.49008287061599717915661941208
