| L(s) = 1 | + 8.42·2-s − 20.3·3-s + 38.9·4-s − 25·5-s − 171.·6-s − 8.66·7-s + 58.6·8-s + 170.·9-s − 210.·10-s − 121·11-s − 791.·12-s + 610.·13-s − 72.9·14-s + 508.·15-s − 752.·16-s + 1.45e3·17-s + 1.43e3·18-s − 361·19-s − 974.·20-s + 175.·21-s − 1.01e3·22-s − 2.84e3·23-s − 1.19e3·24-s + 625·25-s + 5.14e3·26-s + 1.48e3·27-s − 337.·28-s + ⋯ |
| L(s) = 1 | + 1.48·2-s − 1.30·3-s + 1.21·4-s − 0.447·5-s − 1.94·6-s − 0.0668·7-s + 0.324·8-s + 0.699·9-s − 0.665·10-s − 0.301·11-s − 1.58·12-s + 1.00·13-s − 0.0994·14-s + 0.583·15-s − 0.734·16-s + 1.22·17-s + 1.04·18-s − 0.229·19-s − 0.544·20-s + 0.0870·21-s − 0.449·22-s − 1.12·23-s − 0.422·24-s + 0.200·25-s + 1.49·26-s + 0.391·27-s − 0.0813·28-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
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 8.42T + 32T^{2} \) |
| 3 | \( 1 + 20.3T + 243T^{2} \) |
| 7 | \( 1 + 8.66T + 1.68e4T^{2} \) |
| 13 | \( 1 - 610.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.45e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 2.84e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.50e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.44e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.88e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.96e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.12e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 6.63e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.70e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 7.31e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 8.77e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.03e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.32e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.31e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.71e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.87e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.99e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.59e4T + 8.58e9T^{2} \) |
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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
−8.630166763474941844057433867850, −7.65712552460877334908018672751, −6.52831443245821098516787379563, −6.00669824163442681498752104493, −5.36096744240325937764574300794, −4.50036740791622503080911028306, −3.73896164813804440717082358058, −2.74002677918973391188047987093, −1.15965726045491247144566825052, 0,
1.15965726045491247144566825052, 2.74002677918973391188047987093, 3.73896164813804440717082358058, 4.50036740791622503080911028306, 5.36096744240325937764574300794, 6.00669824163442681498752104493, 6.52831443245821098516787379563, 7.65712552460877334908018672751, 8.630166763474941844057433867850
