| L(s) = 1 | + (16.4 + 9.51i)2-s + (80.5 + 8.42i)3-s + (53.1 + 92.0i)4-s + (−46.9 + 27.1i)5-s + (1.24e3 + 905. i)6-s + (−921. + 1.59e3i)7-s − 2.84e3i·8-s + (6.41e3 + 1.35e3i)9-s − 1.03e3·10-s + (−1.58e4 − 9.16e3i)11-s + (3.50e3 + 7.86e3i)12-s + (−1.62e4 − 2.81e4i)13-s + (−3.03e4 + 1.75e4i)14-s + (−4.01e3 + 1.78e3i)15-s + (4.07e4 − 7.05e4i)16-s + 9.75e4i·17-s + ⋯ |
| L(s) = 1 | + (1.03 + 0.594i)2-s + (0.994 + 0.104i)3-s + (0.207 + 0.359i)4-s + (−0.0751 + 0.0434i)5-s + (0.962 + 0.698i)6-s + (−0.383 + 0.664i)7-s − 0.695i·8-s + (0.978 + 0.206i)9-s − 0.103·10-s + (−1.08 − 0.625i)11-s + (0.169 + 0.379i)12-s + (−0.568 − 0.984i)13-s + (−0.790 + 0.456i)14-s + (−0.0792 + 0.0353i)15-s + (0.621 − 1.07i)16-s + 1.16i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.59372 + 0.894533i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.59372 + 0.894533i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-80.5 - 8.42i)T \) |
| good | 2 | \( 1 + (-16.4 - 9.51i)T + (128 + 221. i)T^{2} \) |
| 5 | \( 1 + (46.9 - 27.1i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 7 | \( 1 + (921. - 1.59e3i)T + (-2.88e6 - 4.99e6i)T^{2} \) |
| 11 | \( 1 + (1.58e4 + 9.16e3i)T + (1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 + (1.62e4 + 2.81e4i)T + (-4.07e8 + 7.06e8i)T^{2} \) |
| 17 | \( 1 - 9.75e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 2.08e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + (2.73e5 - 1.58e5i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + (-9.61e4 - 5.55e4i)T + (2.50e11 + 4.33e11i)T^{2} \) |
| 31 | \( 1 + (-1.81e5 - 3.13e5i)T + (-4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 - 1.91e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + (1.16e6 - 6.70e5i)T + (3.99e12 - 6.91e12i)T^{2} \) |
| 43 | \( 1 + (-1.49e6 + 2.58e6i)T + (-5.84e12 - 1.01e13i)T^{2} \) |
| 47 | \( 1 + (1.67e6 + 9.68e5i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + 1.03e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (1.10e7 - 6.37e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-6.41e6 + 1.11e7i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (7.00e6 + 1.21e7i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 + 1.72e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.32e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + (3.55e7 - 6.15e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + (2.74e7 + 1.58e7i)T + (1.12e15 + 1.95e15i)T^{2} \) |
| 89 | \( 1 - 4.71e6iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (-3.45e7 + 5.98e7i)T + (-3.91e15 - 6.78e15i)T^{2} \) |
| 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
−19.71272505244721651447323693097, −18.43212031164162305006876839846, −15.88327899161898524173895891383, −15.18352183957789937216212107620, −13.76451901705363766514836439183, −12.69090407084783591119396253260, −9.879724867134267294090503530760, −7.81289304195674165830379605807, −5.52847757321289050167741889985, −3.26134991249582470067155627626,
2.62538156552917039955036203136, 4.43767167242911547690337494440, 7.59771711347924131217062266456, 9.835050265453939542479426297550, 12.03874739654265038335482482781, 13.44234916846700062455842633091, 14.26310933256917493331357004347, 16.01971187486638441486732040068, 18.22473539799800836114384479023, 19.97954146436456178558270892409
