| L(s) = 1 | − 5.19·2-s + 8.38·3-s + 18.9·4-s + 5·5-s − 43.5·6-s − 4.76·7-s − 57.0·8-s + 43.3·9-s − 25.9·10-s + 11·11-s + 159.·12-s − 70.7·13-s + 24.7·14-s + 41.9·15-s + 144.·16-s − 99.7·17-s − 225.·18-s + 19·19-s + 94.9·20-s − 39.9·21-s − 57.1·22-s − 0.0718·23-s − 478.·24-s + 25·25-s + 367.·26-s + 137.·27-s − 90.3·28-s + ⋯ |
| L(s) = 1 | − 1.83·2-s + 1.61·3-s + 2.37·4-s + 0.447·5-s − 2.96·6-s − 0.257·7-s − 2.52·8-s + 1.60·9-s − 0.821·10-s + 0.301·11-s + 3.82·12-s − 1.50·13-s + 0.472·14-s + 0.721·15-s + 2.25·16-s − 1.42·17-s − 2.94·18-s + 0.229·19-s + 1.06·20-s − 0.414·21-s − 0.553·22-s − 0.000651·23-s − 4.06·24-s + 0.200·25-s + 2.77·26-s + 0.978·27-s − 0.609·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 + 5.19T + 8T^{2} \) |
| 3 | \( 1 - 8.38T + 27T^{2} \) |
| 7 | \( 1 + 4.76T + 343T^{2} \) |
| 13 | \( 1 + 70.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 99.7T + 4.91e3T^{2} \) |
| 23 | \( 1 + 0.0718T + 1.21e4T^{2} \) |
| 29 | \( 1 + 102.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 210.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 10.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 249.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 105.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 167.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 442.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 339.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 116.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 9.99e2T + 3.00e5T^{2} \) |
| 71 | \( 1 + 465.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 207.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 597.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 614.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 405.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 890.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.295098652044884285600473158346, −8.463182194391407017439861528636, −7.75067096943114741670305425654, −7.08820850424962511728226443545, −6.29923698357744818848895441125, −4.55643051785819759865678970444, −3.02576548781218929395543638745, −2.38300265218047762097619716413, −1.57249998701406809563623814150, 0,
1.57249998701406809563623814150, 2.38300265218047762097619716413, 3.02576548781218929395543638745, 4.55643051785819759865678970444, 6.29923698357744818848895441125, 7.08820850424962511728226443545, 7.75067096943114741670305425654, 8.463182194391407017439861528636, 9.295098652044884285600473158346
