| L(s) = 1 | + (−0.783 − 0.324i)2-s + (−0.318 − 1.59i)3-s + (−2.32 − 2.32i)4-s + (5.68 + 3.79i)5-s + (−0.269 + 1.35i)6-s + (−0.751 + 0.502i)7-s + (2.36 + 5.70i)8-s + (5.85 − 2.42i)9-s + (−3.21 − 4.81i)10-s + (6.59 + 1.31i)11-s + (−2.97 + 4.44i)12-s + (−10.5 + 10.5i)13-s + (0.751 − 0.149i)14-s + (4.26 − 10.2i)15-s + 7.89i·16-s + ⋯ |
| L(s) = 1 | + (−0.391 − 0.162i)2-s + (−0.106 − 0.533i)3-s + (−0.580 − 0.580i)4-s + (1.13 + 0.759i)5-s + (−0.0449 + 0.225i)6-s + (−0.107 + 0.0717i)7-s + (0.295 + 0.712i)8-s + (0.650 − 0.269i)9-s + (−0.321 − 0.481i)10-s + (0.599 + 0.119i)11-s + (−0.247 + 0.370i)12-s + (−0.812 + 0.812i)13-s + (0.0536 − 0.0106i)14-s + (0.284 − 0.686i)15-s + 0.493i·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.39953 - 0.365353i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.39953 - 0.365353i\) |
| \(L(2)\) |
|
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 | \( 1 + (0.783 + 0.324i)T + (2.82 + 2.82i)T^{2} \) |
| 3 | \( 1 + (0.318 + 1.59i)T + (-8.31 + 3.44i)T^{2} \) |
| 5 | \( 1 + (-5.68 - 3.79i)T + (9.56 + 23.0i)T^{2} \) |
| 7 | \( 1 + (0.751 - 0.502i)T + (18.7 - 45.2i)T^{2} \) |
| 11 | \( 1 + (-6.59 - 1.31i)T + (111. + 46.3i)T^{2} \) |
| 13 | \( 1 + (10.5 - 10.5i)T - 169iT^{2} \) |
| 19 | \( 1 + (-31.3 - 12.9i)T + (255. + 255. i)T^{2} \) |
| 23 | \( 1 + (-3.63 + 18.2i)T + (-488. - 202. i)T^{2} \) |
| 29 | \( 1 + (-11.7 + 17.6i)T + (-321. - 776. i)T^{2} \) |
| 31 | \( 1 + (-37.3 + 7.42i)T + (887. - 367. i)T^{2} \) |
| 37 | \( 1 + (7.81 + 39.3i)T + (-1.26e3 + 523. i)T^{2} \) |
| 41 | \( 1 + (-3.13 + 2.09i)T + (643. - 1.55e3i)T^{2} \) |
| 43 | \( 1 + (-12.5 + 5.21i)T + (1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (-9.20 + 9.20i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (1.84 + 0.763i)T + (1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-13.1 - 31.7i)T + (-2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-19.7 - 29.5i)T + (-1.42e3 + 3.43e3i)T^{2} \) |
| 67 | \( 1 - 31.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-20.2 - 101. i)T + (-4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (52.3 + 34.9i)T + (2.03e3 + 4.92e3i)T^{2} \) |
| 79 | \( 1 + (64.0 + 12.7i)T + (5.76e3 + 2.38e3i)T^{2} \) |
| 83 | \( 1 + (16.4 - 39.7i)T + (-4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (49.5 + 49.5i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (-70.9 + 106. i)T + (-3.60e3 - 8.69e3i)T^{2} \) |
| 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
−11.47979376277572710682486526988, −10.06122842485926109608596709668, −9.909457664025960857616110467945, −8.974847944041162425474902580199, −7.48335675123512926118293478435, −6.54577358466416443581185853191, −5.70207486233884693300520340691, −4.33917755967940585526683501277, −2.38774651936733110216910938178, −1.20391196650816125005956644444,
1.14538616994240457538042660601, 3.21825011495389430535842807833, 4.71273053824263882484050984516, 5.31253895350184787725210732639, 6.90100412741889699833498428892, 7.932837729221300033477059693072, 9.104131529061521047632550552447, 9.675966126139621876592773598744, 10.21055644172759664747987175172, 11.75428685986243745953151955959
