| L(s) = 1 | − 1.93·2-s + 3·3-s − 4.25·4-s − 18.7·5-s − 5.80·6-s + 6.86·7-s + 23.7·8-s + 9·9-s + 36.3·10-s + 11·11-s − 12.7·12-s + 59.3·13-s − 13.2·14-s − 56.2·15-s − 11.8·16-s + 23.8·17-s − 17.4·18-s + 127.·19-s + 79.8·20-s + 20.5·21-s − 21.2·22-s + 113.·23-s + 71.1·24-s + 226.·25-s − 114.·26-s + 27·27-s − 29.2·28-s + ⋯ |
| L(s) = 1 | − 0.684·2-s + 0.577·3-s − 0.531·4-s − 1.67·5-s − 0.395·6-s + 0.370·7-s + 1.04·8-s + 0.333·9-s + 1.14·10-s + 0.301·11-s − 0.307·12-s + 1.26·13-s − 0.253·14-s − 0.968·15-s − 0.185·16-s + 0.339·17-s − 0.228·18-s + 1.54·19-s + 0.892·20-s + 0.214·21-s − 0.206·22-s + 1.02·23-s + 0.605·24-s + 1.81·25-s − 0.865·26-s + 0.192·27-s − 0.197·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.430399993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.430399993\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 + 1.93T + 8T^{2} \) |
| 5 | \( 1 + 18.7T + 125T^{2} \) |
| 7 | \( 1 - 6.86T + 343T^{2} \) |
| 13 | \( 1 - 59.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 23.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 127.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 113.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 102.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 212.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 202.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 434.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 99.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 138.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 23.4T + 1.48e5T^{2} \) |
| 59 | \( 1 - 45.0T + 2.05e5T^{2} \) |
| 67 | \( 1 + 142.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 730.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 51.4T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.02e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 287.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 658.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 862.T + 9.12e5T^{2} \) |
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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
−8.823860592182653274666724549870, −8.023283105427808533935126522931, −7.60759568802578607736352734444, −6.90775868925406466423355082318, −5.41908485297280123928293665615, −4.52677682311927948914896676508, −3.74856429063294330061556352013, −3.19233418467056212589312652912, −1.38406275054387073611482831783, −0.68765111832588743429529712119,
0.68765111832588743429529712119, 1.38406275054387073611482831783, 3.19233418467056212589312652912, 3.74856429063294330061556352013, 4.52677682311927948914896676508, 5.41908485297280123928293665615, 6.90775868925406466423355082318, 7.60759568802578607736352734444, 8.023283105427808533935126522931, 8.823860592182653274666724549870
