| L(s) = 1 | − 4.91·2-s + 9·3-s − 7.85·4-s − 77.7·5-s − 44.2·6-s − 96.5·7-s + 195.·8-s + 81·9-s + 382.·10-s − 65.0·11-s − 70.6·12-s + 74.4·13-s + 474.·14-s − 700.·15-s − 711.·16-s + 805.·17-s − 398.·18-s + 2.41e3·19-s + 610.·20-s − 869.·21-s + 319.·22-s + 2.62e3·23-s + 1.76e3·24-s + 2.92e3·25-s − 365.·26-s + 729·27-s + 758.·28-s + ⋯ |
| L(s) = 1 | − 0.868·2-s + 0.577·3-s − 0.245·4-s − 1.39·5-s − 0.501·6-s − 0.744·7-s + 1.08·8-s + 0.333·9-s + 1.20·10-s − 0.162·11-s − 0.141·12-s + 0.122·13-s + 0.647·14-s − 0.803·15-s − 0.694·16-s + 0.675·17-s − 0.289·18-s + 1.53·19-s + 0.341·20-s − 0.430·21-s + 0.140·22-s + 1.03·23-s + 0.624·24-s + 0.936·25-s − 0.106·26-s + 0.192·27-s + 0.182·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 9T \) |
| 103 | \( 1 - 1.06e4T \) |
| good | 2 | \( 1 + 4.91T + 32T^{2} \) |
| 5 | \( 1 + 77.7T + 3.12e3T^{2} \) |
| 7 | \( 1 + 96.5T + 1.68e4T^{2} \) |
| 11 | \( 1 + 65.0T + 1.61e5T^{2} \) |
| 13 | \( 1 - 74.4T + 3.71e5T^{2} \) |
| 17 | \( 1 - 805.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.41e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.62e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.53e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.17e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.43e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.36e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.27e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.35e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.10e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 8.18e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.12e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.48e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.84e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.35e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.87e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.62e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.06e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.99e4T + 8.58e9T^{2} \) |
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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
−10.10998342119866098669496456522, −9.347918589733299820571148470378, −8.499642174356985932385123308889, −7.64847902664576593546415029269, −7.08993872892201684149989237616, −5.21827397381619834752593795453, −3.92992597649393183573108612087, −3.12209390823512546719047435387, −1.13622311947857340104221538636, 0,
1.13622311947857340104221538636, 3.12209390823512546719047435387, 3.92992597649393183573108612087, 5.21827397381619834752593795453, 7.08993872892201684149989237616, 7.64847902664576593546415029269, 8.499642174356985932385123308889, 9.347918589733299820571148470378, 10.10998342119866098669496456522
