| L(s) = 1 | − 0.00353·2-s + 4.42·3-s − 31.9·4-s − 0.0156·6-s + 78.1·7-s + 0.226·8-s − 223.·9-s + 343.·11-s − 141.·12-s + 1.05e3·13-s − 0.276·14-s + 1.02e3·16-s + 1.46e3·17-s + 0.790·18-s + 1.05e3·19-s + 346.·21-s − 1.21·22-s − 1.49e3·23-s + 1.00·24-s − 3.73·26-s − 2.06e3·27-s − 2.50e3·28-s + 3.07e3·29-s + 9.43e3·31-s − 10.8·32-s + 1.52e3·33-s − 5.19·34-s + ⋯ |
| L(s) = 1 | − 0.000625·2-s + 0.283·3-s − 0.999·4-s − 0.000177·6-s + 0.603·7-s + 0.00125·8-s − 0.919·9-s + 0.855·11-s − 0.283·12-s + 1.73·13-s − 0.000377·14-s + 0.999·16-s + 1.23·17-s + 0.000574·18-s + 0.671·19-s + 0.171·21-s − 0.000535·22-s − 0.588·23-s + 0.000355·24-s − 0.00108·26-s − 0.544·27-s − 0.603·28-s + 0.678·29-s + 1.76·31-s − 0.00187·32-s + 0.243·33-s − 0.000770·34-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
| \(L(3)\) |
\(\approx\) |
\(2.549890672\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.549890672\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 0.00353T + 32T^{2} \) |
| 3 | \( 1 - 4.42T + 243T^{2} \) |
| 7 | \( 1 - 78.1T + 1.68e4T^{2} \) |
| 11 | \( 1 - 343.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.05e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.46e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.05e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.49e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.07e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.43e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.63e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.80e4T + 1.15e8T^{2} \) |
| 47 | \( 1 - 1.26e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 8.70e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.73e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.15e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 8.88e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.15e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.56e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.22e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.04e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.63e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.40e4T + 8.58e9T^{2} \) |
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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
−8.951711428459484178440278620021, −8.383440651958572055288275463101, −7.929809273197715297332289833044, −6.47243499493044714236992390196, −5.69162725443738020810805072252, −4.85022290760999263673315678751, −3.74759987283171667724962058016, −3.20114519148601414198910646904, −1.51529637456059826545262753587, −0.75712252115976001150682509477,
0.75712252115976001150682509477, 1.51529637456059826545262753587, 3.20114519148601414198910646904, 3.74759987283171667724962058016, 4.85022290760999263673315678751, 5.69162725443738020810805072252, 6.47243499493044714236992390196, 7.929809273197715297332289833044, 8.383440651958572055288275463101, 8.951711428459484178440278620021
