| L(s) = 1 | − 6.49·2-s − 9·3-s + 10.1·4-s + 53.7·5-s + 58.4·6-s − 12.0·7-s + 141.·8-s + 81·9-s − 348.·10-s + 408.·11-s − 91.0·12-s + 189.·13-s + 78.5·14-s − 483.·15-s − 1.24e3·16-s − 1.38e3·17-s − 525.·18-s − 1.11e3·19-s + 543.·20-s + 108.·21-s − 2.65e3·22-s + 514.·23-s − 1.27e3·24-s − 239.·25-s − 1.23e3·26-s − 729·27-s − 122.·28-s + ⋯ |
| L(s) = 1 | − 1.14·2-s − 0.577·3-s + 0.316·4-s + 0.960·5-s + 0.662·6-s − 0.0933·7-s + 0.784·8-s + 0.333·9-s − 1.10·10-s + 1.01·11-s − 0.182·12-s + 0.311·13-s + 0.107·14-s − 0.554·15-s − 1.21·16-s − 1.16·17-s − 0.382·18-s − 0.707·19-s + 0.303·20-s + 0.0538·21-s − 1.16·22-s + 0.202·23-s − 0.452·24-s − 0.0767·25-s − 0.356·26-s − 0.192·27-s − 0.0295·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 9T \) |
| 157 | \( 1 + 2.46e4T \) |
| good | 2 | \( 1 + 6.49T + 32T^{2} \) |
| 5 | \( 1 - 53.7T + 3.12e3T^{2} \) |
| 7 | \( 1 + 12.0T + 1.68e4T^{2} \) |
| 11 | \( 1 - 408.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 189.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.38e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.11e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 514.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.45e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.63e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.49e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 8.73e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.16e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.06e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 7.88e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.81e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.43e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.99e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.10e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.75e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.43e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.12e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.10e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.16e3T + 8.58e9T^{2} \) |
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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
−9.626303482525799322699926898789, −9.076533617642037680618332705063, −8.241670254048605798442779325963, −6.86083991825677683094661107640, −6.37396035705189731765425190603, −5.09888563039442534791156030724, −4.01729878308848391395145868618, −2.11953966260370855861839028408, −1.23163947144076272152493808390, 0,
1.23163947144076272152493808390, 2.11953966260370855861839028408, 4.01729878308848391395145868618, 5.09888563039442534791156030724, 6.37396035705189731765425190603, 6.86083991825677683094661107640, 8.241670254048605798442779325963, 9.076533617642037680618332705063, 9.626303482525799322699926898789
