| L(s) = 1 | + 5.48·2-s + 3.40·3-s − 481.·4-s + 1.65e3·5-s + 18.6·6-s − 5.45e3·8-s − 1.96e4·9-s + 9.09e3·10-s + 3.36e4·11-s − 1.64e3·12-s + 4.79e4·13-s + 5.64e3·15-s + 2.16e5·16-s − 3.45e5·17-s − 1.07e5·18-s − 4.05e5·19-s − 7.98e5·20-s + 1.84e5·22-s − 1.85e6·23-s − 1.85e4·24-s + 7.95e5·25-s + 2.62e5·26-s − 1.34e5·27-s + 6.82e5·29-s + 3.09e4·30-s − 9.09e6·31-s + 3.98e6·32-s + ⋯ |
| L(s) = 1 | + 0.242·2-s + 0.0242·3-s − 0.941·4-s + 1.18·5-s + 0.00588·6-s − 0.470·8-s − 0.999·9-s + 0.287·10-s + 0.693·11-s − 0.0228·12-s + 0.465·13-s + 0.0288·15-s + 0.826·16-s − 1.00·17-s − 0.242·18-s − 0.713·19-s − 1.11·20-s + 0.168·22-s − 1.38·23-s − 0.0114·24-s + 0.407·25-s + 0.112·26-s − 0.0485·27-s + 0.179·29-s + 0.00698·30-s − 1.76·31-s + 0.671·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - 5.48T + 512T^{2} \) |
| 3 | \( 1 - 3.40T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.65e3T + 1.95e6T^{2} \) |
| 11 | \( 1 - 3.36e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 4.79e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.45e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 4.05e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.85e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 6.82e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 9.09e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.55e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.98e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 6.28e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.03e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 6.64e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 7.05e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 8.43e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.11e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.31e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.48e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.66e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.33e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 6.22e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 9.94e8T + 7.60e17T^{2} \) |
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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
−13.43550210067510221895399609158, −12.08944767222168420589979330738, −10.56601610496650262370447527789, −9.272281838284237377841758879983, −8.547619174217333822773274489711, −6.34729338207166407008882748661, −5.37598676122978497991816232051, −3.78409323647739874336066890012, −1.95295451669101750720933445766, 0,
1.95295451669101750720933445766, 3.78409323647739874336066890012, 5.37598676122978497991816232051, 6.34729338207166407008882748661, 8.547619174217333822773274489711, 9.272281838284237377841758879983, 10.56601610496650262370447527789, 12.08944767222168420589979330738, 13.43550210067510221895399609158
