| L(s) = 1 | − 0.650·2-s − 4.85·3-s − 31.5·4-s + 3.15·6-s + 115.·7-s + 41.3·8-s − 219.·9-s + 421.·11-s + 153.·12-s + 611.·13-s − 75.1·14-s + 983.·16-s + 389.·17-s + 142.·18-s + 2.55e3·19-s − 560.·21-s − 274.·22-s − 132.·23-s − 200.·24-s − 397.·26-s + 2.24e3·27-s − 3.64e3·28-s − 2.36e3·29-s + 8.45e3·31-s − 1.96e3·32-s − 2.04e3·33-s − 253.·34-s + ⋯ |
| L(s) = 1 | − 0.114·2-s − 0.311·3-s − 0.986·4-s + 0.0357·6-s + 0.891·7-s + 0.228·8-s − 0.903·9-s + 1.05·11-s + 0.307·12-s + 1.00·13-s − 0.102·14-s + 0.960·16-s + 0.326·17-s + 0.103·18-s + 1.62·19-s − 0.277·21-s − 0.120·22-s − 0.0523·23-s − 0.0710·24-s − 0.115·26-s + 0.592·27-s − 0.879·28-s − 0.522·29-s + 1.57·31-s − 0.338·32-s − 0.326·33-s − 0.0375·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.061077147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.061077147\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 - 1.84e3T \) |
| good | 2 | \( 1 + 0.650T + 32T^{2} \) |
| 3 | \( 1 + 4.85T + 243T^{2} \) |
| 7 | \( 1 - 115.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 421.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 611.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 389.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.55e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 132.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.36e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.45e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.50e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.87e4T + 1.15e8T^{2} \) |
| 47 | \( 1 + 2.84e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.00e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.78e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.29e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.06e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.28e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.85e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.07e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.96e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.23e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.49e5T + 8.58e9T^{2} \) |
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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
−9.247639738624647308021599263936, −8.226800065107852508501302644628, −7.891836739029497964767373157867, −6.45860655985225294023929808922, −5.66331408711159118012430323354, −4.88942526334229972952064994835, −3.99206536924203126051289879379, −3.04009104719604954835740956302, −1.35632757350566385292411014481, −0.75388134452022802233080215885,
0.75388134452022802233080215885, 1.35632757350566385292411014481, 3.04009104719604954835740956302, 3.99206536924203126051289879379, 4.88942526334229972952064994835, 5.66331408711159118012430323354, 6.45860655985225294023929808922, 7.891836739029497964767373157867, 8.226800065107852508501302644628, 9.247639738624647308021599263936
