L(s) = 1 | − 2.73·2-s + 28.3·3-s − 24.5·4-s − 25·5-s − 77.5·6-s + 154.·8-s + 559.·9-s + 68.4·10-s + 63.4·11-s − 694.·12-s + 868.·13-s − 708.·15-s + 361.·16-s − 1.18e3·17-s − 1.53e3·18-s − 1.53e3·19-s + 612.·20-s − 173.·22-s − 2.33e3·23-s + 4.38e3·24-s + 625·25-s − 2.37e3·26-s + 8.98e3·27-s + 8.16e3·29-s + 1.93e3·30-s + 2.36e3·31-s − 5.93e3·32-s + ⋯ |
L(s) = 1 | − 0.483·2-s + 1.81·3-s − 0.765·4-s − 0.447·5-s − 0.879·6-s + 0.854·8-s + 2.30·9-s + 0.216·10-s + 0.158·11-s − 1.39·12-s + 1.42·13-s − 0.812·15-s + 0.352·16-s − 0.998·17-s − 1.11·18-s − 0.974·19-s + 0.342·20-s − 0.0764·22-s − 0.921·23-s + 1.55·24-s + 0.200·25-s − 0.689·26-s + 2.37·27-s + 1.80·29-s + 0.393·30-s + 0.442·31-s − 1.02·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.559770708\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.559770708\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + 25T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 2.73T + 32T^{2} \) |
| 3 | \( 1 - 28.3T + 243T^{2} \) |
| 11 | \( 1 - 63.4T + 1.61e5T^{2} \) |
| 13 | \( 1 - 868.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.18e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.53e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.33e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.16e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.36e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.28e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.65e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.29e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 6.21e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.99e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.07e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.99e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.16e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.27e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.44e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.76e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.65e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.53e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.80e4T + 8.58e9T^{2} \) |
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
−10.88574458126788801555100890152, −9.949798936948476559314324594910, −8.945219934309692237807970118647, −8.463383382956362943025038399728, −7.85707616644752979464692889716, −6.51255118783850308029936439725, −4.37043760681561692996325066244, −3.84900572837529301844258641920, −2.41348004085787319856224003697, −0.993567703977594493397737669528,
0.993567703977594493397737669528, 2.41348004085787319856224003697, 3.84900572837529301844258641920, 4.37043760681561692996325066244, 6.51255118783850308029936439725, 7.85707616644752979464692889716, 8.463383382956362943025038399728, 8.945219934309692237807970118647, 9.949798936948476559314324594910, 10.88574458126788801555100890152