| L(s) = 1 | + 7.37·2-s − 3.41·3-s + 22.4·4-s + 108.·5-s − 25.1·6-s − 80.6·7-s − 70.6·8-s − 231.·9-s + 800.·10-s − 472.·11-s − 76.5·12-s + 639.·13-s − 594.·14-s − 370.·15-s − 1.23e3·16-s + 478.·17-s − 1.70e3·18-s + 361·19-s + 2.43e3·20-s + 275.·21-s − 3.48e3·22-s − 190.·23-s + 241.·24-s + 8.65e3·25-s + 4.71e3·26-s + 1.61e3·27-s − 1.80e3·28-s + ⋯ |
| L(s) = 1 | + 1.30·2-s − 0.219·3-s + 0.700·4-s + 1.94·5-s − 0.285·6-s − 0.622·7-s − 0.390·8-s − 0.952·9-s + 2.53·10-s − 1.17·11-s − 0.153·12-s + 1.04·13-s − 0.811·14-s − 0.425·15-s − 1.20·16-s + 0.401·17-s − 1.24·18-s + 0.229·19-s + 1.35·20-s + 0.136·21-s − 1.53·22-s − 0.0749·23-s + 0.0855·24-s + 2.76·25-s + 1.36·26-s + 0.427·27-s − 0.435·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.519319203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.519319203\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 - 361T \) |
| good | 2 | \( 1 - 7.37T + 32T^{2} \) |
| 3 | \( 1 + 3.41T + 243T^{2} \) |
| 5 | \( 1 - 108.T + 3.12e3T^{2} \) |
| 7 | \( 1 + 80.6T + 1.68e4T^{2} \) |
| 11 | \( 1 + 472.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 639.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 478.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 190.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.84e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 381.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.53e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.30e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.07e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.16e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 6.50e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.94e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.09e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.13e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.83e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 829.T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.82e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.61e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.14e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.63e5T + 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.45498751107657677016098092274, −15.99503224744896544461783648807, −14.30125195043826012973489098203, −13.54321105546753378491555252021, −12.67648076189829934273544966978, −10.72770315762460556906986048843, −9.175710294921058131929316016309, −6.12505511149204918988044270822, −5.45686701406109831448983355719, −2.79576152434414397203283737437,
2.79576152434414397203283737437, 5.45686701406109831448983355719, 6.12505511149204918988044270822, 9.175710294921058131929316016309, 10.72770315762460556906986048843, 12.67648076189829934273544966978, 13.54321105546753378491555252021, 14.30125195043826012973489098203, 15.99503224744896544461783648807, 17.45498751107657677016098092274
