| L(s) = 1 | + 2.56·2-s − 1.43·4-s + 5·5-s + 6.24·7-s − 24.1·8-s + 12.8·10-s + 11·11-s − 49.1·13-s + 16·14-s − 50.4·16-s − 82.7·17-s − 130.·19-s − 7.19·20-s + 28.1·22-s + 185.·23-s + 25·25-s − 125.·26-s − 8.98·28-s + 8.90·29-s + 5.26·31-s + 64.2·32-s − 211.·34-s + 31.2·35-s − 416.·37-s − 333.·38-s − 120.·40-s + 298.·41-s + ⋯ |
| L(s) = 1 | + 0.905·2-s − 0.179·4-s + 0.447·5-s + 0.337·7-s − 1.06·8-s + 0.405·10-s + 0.301·11-s − 1.04·13-s + 0.305·14-s − 0.787·16-s − 1.17·17-s − 1.57·19-s − 0.0804·20-s + 0.273·22-s + 1.68·23-s + 0.200·25-s − 0.949·26-s − 0.0606·28-s + 0.0570·29-s + 0.0304·31-s + 0.354·32-s − 1.06·34-s + 0.150·35-s − 1.85·37-s − 1.42·38-s − 0.477·40-s + 1.13·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 2.56T + 8T^{2} \) |
| 7 | \( 1 - 6.24T + 343T^{2} \) |
| 13 | \( 1 + 49.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 82.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 130.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 185.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 8.90T + 2.43e4T^{2} \) |
| 31 | \( 1 - 5.26T + 2.97e4T^{2} \) |
| 37 | \( 1 + 416.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 298.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 513.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 557.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 168.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 618.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 786.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 339.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.12e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 123.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 309.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.02e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 141.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 798.T + 9.12e5T^{2} \) |
| 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
−10.13245662428900951426411017863, −9.077248386688530555116035978633, −8.527017468236425571268673193362, −7.00130698532024725838238632290, −6.27472499112418587661656578341, −5.00070831600848173878349309545, −4.57726716946648769513464080531, −3.21493357478241499562303704031, −1.99804556965075284609601186932, 0,
1.99804556965075284609601186932, 3.21493357478241499562303704031, 4.57726716946648769513464080531, 5.00070831600848173878349309545, 6.27472499112418587661656578341, 7.00130698532024725838238632290, 8.527017468236425571268673193362, 9.077248386688530555116035978633, 10.13245662428900951426411017863
