| L(s) = 1 | − 9·3-s + 46.1·5-s + 81·9-s − 631.·11-s − 1.07e3·13-s − 415.·15-s + 161.·17-s + 1.17e3·19-s + 2.16e3·23-s − 998.·25-s − 729·27-s − 4.49e3·29-s + 318.·31-s + 5.68e3·33-s + 1.51e4·37-s + 9.71e3·39-s + 2.05e4·41-s − 455.·43-s + 3.73e3·45-s − 2.07e4·47-s − 1.45e3·51-s − 1.93e4·53-s − 2.91e4·55-s − 1.05e4·57-s + 6.36e3·59-s − 4.91e4·61-s − 4.97e4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.824·5-s + 0.333·9-s − 1.57·11-s − 1.77·13-s − 0.476·15-s + 0.135·17-s + 0.747·19-s + 0.852·23-s − 0.319·25-s − 0.192·27-s − 0.991·29-s + 0.0594·31-s + 0.908·33-s + 1.82·37-s + 1.02·39-s + 1.91·41-s − 0.0375·43-s + 0.274·45-s − 1.37·47-s − 0.0780·51-s − 0.943·53-s − 1.29·55-s − 0.431·57-s + 0.238·59-s − 1.69·61-s − 1.46·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.356980220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.356980220\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 46.1T + 3.12e3T^{2} \) |
| 11 | \( 1 + 631.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.07e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 161.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.17e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.16e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.49e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 318.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.51e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.05e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 455.T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.07e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.93e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 6.36e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.91e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.40e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.29e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.86e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.44e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.04e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.02e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.40e4T + 8.58e9T^{2} \) |
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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
−9.724223006547540992327786473008, −9.540564798091820657595075232742, −7.84431425494172624980308669078, −7.37270635686067023315511277723, −6.09756040032114426162042649551, −5.30618578983063020268865576510, −4.68324149007996098994271096569, −2.93010157898980910543318897937, −2.07640048635663732507483068213, −0.56103016638208437468744430182,
0.56103016638208437468744430182, 2.07640048635663732507483068213, 2.93010157898980910543318897937, 4.68324149007996098994271096569, 5.30618578983063020268865576510, 6.09756040032114426162042649551, 7.37270635686067023315511277723, 7.84431425494172624980308669078, 9.540564798091820657595075232742, 9.724223006547540992327786473008
