| L(s) = 1 | + 7.54·2-s + 24.9·4-s − 87.4·5-s − 41.2·7-s − 52.9·8-s − 660.·10-s − 7.68·11-s − 605.·13-s − 311.·14-s − 1.19e3·16-s − 566.·17-s + 2.06e3·19-s − 2.18e3·20-s − 58.0·22-s − 274.·23-s + 4.51e3·25-s − 4.56e3·26-s − 1.02e3·28-s − 4.82e3·29-s + 4.46e3·31-s − 7.35e3·32-s − 4.27e3·34-s + 3.60e3·35-s + 242.·37-s + 1.55e4·38-s + 4.62e3·40-s − 3.50e3·41-s + ⋯ |
| L(s) = 1 | + 1.33·2-s + 0.781·4-s − 1.56·5-s − 0.317·7-s − 0.292·8-s − 2.08·10-s − 0.0191·11-s − 0.993·13-s − 0.424·14-s − 1.17·16-s − 0.475·17-s + 1.30·19-s − 1.22·20-s − 0.0255·22-s − 0.108·23-s + 1.44·25-s − 1.32·26-s − 0.248·28-s − 1.06·29-s + 0.833·31-s − 1.27·32-s − 0.634·34-s + 0.497·35-s + 0.0291·37-s + 1.74·38-s + 0.457·40-s − 0.326·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 2 | \( 1 - 7.54T + 32T^{2} \) |
| 5 | \( 1 + 87.4T + 3.12e3T^{2} \) |
| 7 | \( 1 + 41.2T + 1.68e4T^{2} \) |
| 11 | \( 1 + 7.68T + 1.61e5T^{2} \) |
| 13 | \( 1 + 605.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 566.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.06e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 274.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.82e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.46e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 242.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.50e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.53e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.79e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.02e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.01e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.83e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.18e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.26e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.28e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.81e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.05e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.24e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.62e5T + 8.58e9T^{2} \) |
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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
−12.75837576512546558505169562211, −11.97896496588734954748431642893, −11.24842328118685749490836546936, −9.479923214122250271905649566485, −7.921242555067643986472447417922, −6.81372253274253398346755417418, −5.17695417311365833691316772244, −4.09367036710691220640070084621, −3.00709668381469799242400735515, 0,
3.00709668381469799242400735515, 4.09367036710691220640070084621, 5.17695417311365833691316772244, 6.81372253274253398346755417418, 7.921242555067643986472447417922, 9.479923214122250271905649566485, 11.24842328118685749490836546936, 11.97896496588734954748431642893, 12.75837576512546558505169562211
