| L(s) = 1 | − 6.14·2-s + 9·3-s + 5.78·4-s + 64.1·5-s − 55.3·6-s − 49·7-s + 161.·8-s + 81·9-s − 394.·10-s + 14.3·11-s + 52.0·12-s + 275.·13-s + 301.·14-s + 577.·15-s − 1.17e3·16-s + 78.9·17-s − 497.·18-s − 2.60e3·19-s + 371.·20-s − 441·21-s − 87.9·22-s + 529·23-s + 1.45e3·24-s + 991.·25-s − 1.69e3·26-s + 729·27-s − 283.·28-s + ⋯ |
| L(s) = 1 | − 1.08·2-s + 0.577·3-s + 0.180·4-s + 1.14·5-s − 0.627·6-s − 0.377·7-s + 0.890·8-s + 0.333·9-s − 1.24·10-s + 0.0356·11-s + 0.104·12-s + 0.452·13-s + 0.410·14-s + 0.662·15-s − 1.14·16-s + 0.0662·17-s − 0.362·18-s − 1.65·19-s + 0.207·20-s − 0.218·21-s − 0.0387·22-s + 0.208·23-s + 0.513·24-s + 0.317·25-s − 0.491·26-s + 0.192·27-s − 0.0683·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 9T \) |
| 7 | \( 1 + 49T \) |
| 23 | \( 1 - 529T \) |
| good | 2 | \( 1 + 6.14T + 32T^{2} \) |
| 5 | \( 1 - 64.1T + 3.12e3T^{2} \) |
| 11 | \( 1 - 14.3T + 1.61e5T^{2} \) |
| 13 | \( 1 - 275.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 78.9T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.60e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 3.94e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.14e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.64e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.00e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.05e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 114.T + 2.29e8T^{2} \) |
| 53 | \( 1 + 6.23e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.17e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.22e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.50e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.11e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.61e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.33e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.56e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.10e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.49e5T + 8.58e9T^{2} \) |
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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
−9.694471217886779951182395390505, −8.806336152181669041995638525162, −8.384999866866701469712980908939, −7.10540545298574100665040711694, −6.28364925369938801632802486208, −5.01466416664159134015600768729, −3.72193042882390954906635444009, −2.25040458206572099702436470947, −1.45574821237892590960279107501, 0,
1.45574821237892590960279107501, 2.25040458206572099702436470947, 3.72193042882390954906635444009, 5.01466416664159134015600768729, 6.28364925369938801632802486208, 7.10540545298574100665040711694, 8.384999866866701469712980908939, 8.806336152181669041995638525162, 9.694471217886779951182395390505
