| L(s) = 1 | + (−4.89 + 2.82i)2-s + (−11.1 + 24.5i)3-s + (15.9 − 27.7i)4-s + (−39.5 − 22.8i)5-s + (−14.6 − 152. i)6-s + (−245. − 424. i)7-s + 181. i·8-s + (−478. − 550. i)9-s + 258.·10-s + (−873. + 504. i)11-s + (501. + 703. i)12-s + (466. − 808. i)13-s + (2.40e3 + 1.38e3i)14-s + (1.00e3 − 716. i)15-s + (−512. − 886. i)16-s + 8.09e3i·17-s + ⋯ |
| L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.414 + 0.909i)3-s + (0.249 − 0.433i)4-s + (−0.316 − 0.182i)5-s + (−0.0677 − 0.703i)6-s + (−0.714 − 1.23i)7-s + 0.353i·8-s + (−0.655 − 0.754i)9-s + 0.258·10-s + (−0.656 + 0.378i)11-s + (0.290 + 0.407i)12-s + (0.212 − 0.368i)13-s + (0.875 + 0.505i)14-s + (0.297 − 0.212i)15-s + (−0.125 − 0.216i)16-s + 1.64i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 + 0.777i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.629 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0318399 - 0.0667645i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0318399 - 0.0667645i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4.89 - 2.82i)T \) |
| 3 | \( 1 + (11.1 - 24.5i)T \) |
| good | 5 | \( 1 + (39.5 + 22.8i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 7 | \( 1 + (245. + 424. i)T + (-5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (873. - 504. i)T + (8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + (-466. + 808. i)T + (-2.41e6 - 4.18e6i)T^{2} \) |
| 17 | \( 1 - 8.09e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 7.72e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (1.18e4 + 6.84e3i)T + (7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + (1.96e3 - 1.13e3i)T + (2.97e8 - 5.15e8i)T^{2} \) |
| 31 | \( 1 + (1.70e4 - 2.95e4i)T + (-4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 - 9.20e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (3.10e4 + 1.79e4i)T + (2.37e9 + 4.11e9i)T^{2} \) |
| 43 | \( 1 + (-3.45e4 - 5.98e4i)T + (-3.16e9 + 5.47e9i)T^{2} \) |
| 47 | \( 1 + (-1.32e4 + 7.62e3i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + 2.36e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (2.21e5 + 1.28e5i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (1.99e4 + 3.45e4i)T + (-2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (1.60e5 - 2.77e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 - 4.04e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 3.93e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (4.49e5 + 7.78e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-1.54e5 + 8.94e4i)T + (1.63e11 - 2.83e11i)T^{2} \) |
| 89 | \( 1 - 8.26e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (3.17e5 + 5.50e5i)T + (-4.16e11 + 7.21e11i)T^{2} \) |
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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
−16.75939012292823276147856828268, −15.93398224942121299746825444193, −14.69785449599607124064918014746, −12.77161868001907682824024231596, −10.73917007710819511588954036731, −10.04541249643037992735655091509, −8.208698560763989655009045311140, −6.28948962922804108803581722212, −4.11972677104944947336130834320, −0.05974136556983953401149895312,
2.51361554893894988813593547435, 5.99014583345743936276167598944, 7.67850870224326549487971886562, 9.244253685062874394402702866234, 11.20287774412839682591382636092, 12.19402719574377865925444801967, 13.44314677945778322562652032342, 15.54107748701911683921088762785, 16.70336866291349482255243499531, 18.39116073141736199074552062969
