| L(s) = 1 | − 37.5·2-s + 70.9·3-s + 898.·4-s − 98.5·5-s − 2.66e3·6-s − 4.15e3·7-s − 1.44e4·8-s − 1.46e4·9-s + 3.70e3·10-s − 1.76e4·11-s + 6.37e4·12-s + 1.25e5·13-s + 1.56e5·14-s − 6.99e3·15-s + 8.46e4·16-s + 5.50e5·18-s + 1.86e5·19-s − 8.84e4·20-s − 2.95e5·21-s + 6.63e5·22-s − 8.80e5·23-s − 1.02e6·24-s − 1.94e6·25-s − 4.70e6·26-s − 2.43e6·27-s − 3.73e6·28-s − 4.72e5·29-s + ⋯ |
| L(s) = 1 | − 1.65·2-s + 0.505·3-s + 1.75·4-s − 0.0705·5-s − 0.839·6-s − 0.654·7-s − 1.25·8-s − 0.744·9-s + 0.117·10-s − 0.363·11-s + 0.887·12-s + 1.21·13-s + 1.08·14-s − 0.0356·15-s + 0.322·16-s + 1.23·18-s + 0.328·19-s − 0.123·20-s − 0.331·21-s + 0.603·22-s − 0.656·23-s − 0.632·24-s − 0.995·25-s − 2.01·26-s − 0.882·27-s − 1.14·28-s − 0.124·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.5593720675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5593720675\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + 37.5T + 512T^{2} \) |
| 3 | \( 1 - 70.9T + 1.96e4T^{2} \) |
| 5 | \( 1 + 98.5T + 1.95e6T^{2} \) |
| 7 | \( 1 + 4.15e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 1.76e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.25e5T + 1.06e10T^{2} \) |
| 19 | \( 1 - 1.86e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 8.80e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.72e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 5.47e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.99e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.70e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.41e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.99e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.20e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.43e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 9.63e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.54e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.43e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.81e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.31e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 3.37e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.13e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 5.74e8T + 7.60e17T^{2} \) |
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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
−9.814337982371865016017210156745, −9.352518382558982598290097828968, −8.286748006821261641720767401561, −7.898219203407406043331957641273, −6.63609274599177930237990416568, −5.72957928734580310943501100679, −3.77496850058422733532173694121, −2.71065947485846860156591561408, −1.64614391423550409035889930545, −0.41281970852379811783261553037,
0.41281970852379811783261553037, 1.64614391423550409035889930545, 2.71065947485846860156591561408, 3.77496850058422733532173694121, 5.72957928734580310943501100679, 6.63609274599177930237990416568, 7.898219203407406043331957641273, 8.286748006821261641720767401561, 9.352518382558982598290097828968, 9.814337982371865016017210156745
