| L(s) = 1 | + (−0.938 + 1.05i)2-s + (−0.0518 + 1.73i)3-s + (−0.239 − 1.98i)4-s + (−1.55 + 1.84i)5-s + (−1.78 − 1.67i)6-s + (−1.49 − 0.544i)7-s + (2.32 + 1.60i)8-s + (−2.99 − 0.179i)9-s + (−0.500 − 3.37i)10-s + (0.102 + 0.122i)11-s + (3.45 − 0.311i)12-s + (−3.39 − 0.597i)13-s + (1.98 − 1.07i)14-s + (−3.12 − 2.78i)15-s + (−3.88 + 0.950i)16-s + (−1.43 + 2.48i)17-s + ⋯ |
| L(s) = 1 | + (−0.663 + 0.748i)2-s + (−0.0299 + 0.999i)3-s + (−0.119 − 0.992i)4-s + (−0.693 + 0.826i)5-s + (−0.728 − 0.685i)6-s + (−0.565 − 0.205i)7-s + (0.822 + 0.569i)8-s + (−0.998 − 0.0597i)9-s + (−0.158 − 1.06i)10-s + (0.0308 + 0.0367i)11-s + (0.995 − 0.0898i)12-s + (−0.940 − 0.165i)13-s + (0.529 − 0.286i)14-s + (−0.805 − 0.718i)15-s + (−0.971 + 0.237i)16-s + (−0.348 + 0.603i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.833 + 0.551i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.833 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.112108 - 0.372510i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.112108 - 0.372510i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.938 - 1.05i)T \) |
| 3 | \( 1 + (0.0518 - 1.73i)T \) |
| good | 5 | \( 1 + (1.55 - 1.84i)T + (-0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (1.49 + 0.544i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (-0.102 - 0.122i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (3.39 + 0.597i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (1.43 - 2.48i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.75 + 2.16i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.14 + 0.781i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (8.60 - 1.51i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (6.69 - 2.43i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-5.14 - 2.97i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.74 - 9.90i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-7.59 - 9.05i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-3.66 - 1.33i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 - 9.80iT - 53T^{2} \) |
| 59 | \( 1 + (1.72 - 2.05i)T + (-10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.492 + 1.35i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (0.225 + 0.0397i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-2.82 + 4.88i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.14 - 12.3i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.18 + 12.3i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (11.1 - 1.95i)T + (77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (3.73 + 6.46i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.17 + 4.34i)T + (16.8 - 95.5i)T^{2} \) |
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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
−12.99406783444204343934447017131, −11.36165852414000258821274238645, −10.85102478348525792205044540635, −9.788496146836310465329484966603, −9.170720661710757342041952460003, −7.80954285448698742886416142668, −7.00031668978155989878203886771, −5.76908831453123610202749786410, −4.50445058463553883262690427762, −3.08361276192077184171137013195,
0.39258460730639900100475068046, 2.22834829117737302936665440582, 3.74297121169264400393574392135, 5.38955987979508479022044334035, 7.15337570075540553438482676274, 7.71303867893534249085415312438, 8.906888658488463145670669868159, 9.523941722662528485031776081183, 11.06561510973467051739253343814, 11.88302784750431898536810645863
