L(s) = 1 | + (0.965 − 0.258i)2-s + (1.61 + 0.638i)3-s + (0.866 − 0.499i)4-s + (−0.489 − 0.489i)5-s + (1.72 + 0.199i)6-s + (0.258 − 0.965i)7-s + (0.707 − 0.707i)8-s + (2.18 + 2.05i)9-s + (−0.599 − 0.345i)10-s + (0.247 + 0.923i)11-s + (1.71 − 0.252i)12-s + (2.67 + 2.41i)13-s − i·14-s + (−0.475 − 1.09i)15-s + (0.500 − 0.866i)16-s + (−1.15 − 2.00i)17-s + ⋯ |
L(s) = 1 | + (0.683 − 0.183i)2-s + (0.929 + 0.368i)3-s + (0.433 − 0.249i)4-s + (−0.218 − 0.218i)5-s + (0.702 + 0.0815i)6-s + (0.0978 − 0.365i)7-s + (0.249 − 0.249i)8-s + (0.728 + 0.685i)9-s + (−0.189 − 0.109i)10-s + (0.0745 + 0.278i)11-s + (0.494 − 0.0728i)12-s + (0.742 + 0.669i)13-s − 0.267i·14-s + (−0.122 − 0.283i)15-s + (0.125 − 0.216i)16-s + (−0.280 − 0.485i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.102i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.87404 - 0.147939i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.87404 - 0.147939i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 + 0.258i)T \) |
| 3 | \( 1 + (-1.61 - 0.638i)T \) |
| 7 | \( 1 + (-0.258 + 0.965i)T \) |
| 13 | \( 1 + (-2.67 - 2.41i)T \) |
good | 5 | \( 1 + (0.489 + 0.489i)T + 5iT^{2} \) |
| 11 | \( 1 + (-0.247 - 0.923i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (1.15 + 2.00i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.656 + 0.175i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (0.227 - 0.393i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.11 + 2.37i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.27 + 1.27i)T - 31iT^{2} \) |
| 37 | \( 1 + (-1.67 + 0.447i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (9.15 - 2.45i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-0.759 + 0.438i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (7.33 - 7.33i)T - 47iT^{2} \) |
| 53 | \( 1 - 10.0iT - 53T^{2} \) |
| 59 | \( 1 + (11.3 + 3.04i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-6.88 - 11.9i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.53 + 9.44i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-0.951 + 3.55i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (4.44 + 4.44i)T + 73iT^{2} \) |
| 79 | \( 1 + 3.18T + 79T^{2} \) |
| 83 | \( 1 + (0.476 + 0.476i)T + 83iT^{2} \) |
| 89 | \( 1 + (0.780 + 2.91i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (11.6 + 3.12i)T + (84.0 + 48.5i)T^{2} \) |
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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
−10.83760549930628311966494166171, −9.905670714863189875434162219856, −9.075764192607969254278788579526, −8.128458510163010164263381341341, −7.22708199404724083753282696212, −6.17561170932159867855287949080, −4.70172459333197693142075911865, −4.15281630115718182372204747344, −3.04341649001600749061865186038, −1.73241137474838899826155329687,
1.76450025851933982231380194474, 3.13960513242232572719791431880, 3.81724602992770694311468098785, 5.22876360975350098659322530719, 6.32163505219106122805630400996, 7.16614354226553655174440708205, 8.192121846160436929406618595820, 8.713980253781043258401334647812, 9.916184631757900521126154632964, 10.97501998166623818692742411846