| L(s) = 1 | + 1.26·2-s + 3.55·3-s − 30.3·4-s − 25·5-s + 4.49·6-s + 122.·7-s − 78.9·8-s − 230.·9-s − 31.6·10-s + 121·11-s − 107.·12-s + 885.·13-s + 154.·14-s − 88.7·15-s + 872.·16-s + 1.02e3·17-s − 291.·18-s − 361·19-s + 759.·20-s + 433.·21-s + 153.·22-s − 913.·23-s − 280.·24-s + 625·25-s + 1.11e3·26-s − 1.68e3·27-s − 3.71e3·28-s + ⋯ |
| L(s) = 1 | + 0.223·2-s + 0.227·3-s − 0.949·4-s − 0.447·5-s + 0.0509·6-s + 0.941·7-s − 0.436·8-s − 0.948·9-s − 0.100·10-s + 0.301·11-s − 0.216·12-s + 1.45·13-s + 0.210·14-s − 0.101·15-s + 0.852·16-s + 0.856·17-s − 0.212·18-s − 0.229·19-s + 0.424·20-s + 0.214·21-s + 0.0674·22-s − 0.359·23-s − 0.0993·24-s + 0.200·25-s + 0.324·26-s − 0.443·27-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
| \(L(3)\) |
\(\approx\) |
\(1.968717937\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.968717937\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 1.26T + 32T^{2} \) |
| 3 | \( 1 - 3.55T + 243T^{2} \) |
| 7 | \( 1 - 122.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 885.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.02e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 913.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.74e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.28e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.68e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.51e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.50e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.20e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.00e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.24e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.34e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.98e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.52e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.72e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.80e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.01e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.33e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.45e4T + 8.58e9T^{2} \) |
| 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
−8.933213215265858433702217658141, −8.303314333176655368294733995825, −7.993612634309575149222447910254, −6.53426080049633722275678586072, −5.58178595556708323110353104775, −4.87623212115304185364641587247, −3.80676896586175427020307631219, −3.25742297826583700623283802632, −1.66829587138837942738340246207, −0.60731824893073960075670754493,
0.60731824893073960075670754493, 1.66829587138837942738340246207, 3.25742297826583700623283802632, 3.80676896586175427020307631219, 4.87623212115304185364641587247, 5.58178595556708323110353104775, 6.53426080049633722275678586072, 7.993612634309575149222447910254, 8.303314333176655368294733995825, 8.933213215265858433702217658141
