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
−10.97501998166623818692742411846, −9.916184631757900521126154632964, −8.713980253781043258401334647812, −8.192121846160436929406618595820, −7.16614354226553655174440708205, −6.32163505219106122805630400996, −5.22876360975350098659322530719, −3.81724602992770694311468098785, −3.13960513242232572719791431880, −1.76450025851933982231380194474,
1.73241137474838899826155329687, 3.04341649001600749061865186038, 4.15281630115718182372204747344, 4.70172459333197693142075911865, 6.17561170932159867855287949080, 7.22708199404724083753282696212, 8.128458510163010164263381341341, 9.075764192607969254278788579526, 9.905670714863189875434162219856, 10.83760549930628311966494166171