| L(s) = 1 | + (−0.841 − 0.540i)2-s + (−0.142 − 0.989i)3-s + (0.415 + 0.909i)4-s + (−0.180 − 0.0530i)5-s + (−0.415 + 0.909i)6-s + (−0.654 + 0.755i)7-s + (0.142 − 0.989i)8-s + (−0.959 + 0.281i)9-s + (0.123 + 0.142i)10-s + (3.80 − 2.44i)11-s + (0.841 − 0.540i)12-s + (0.793 + 0.915i)13-s + (0.959 − 0.281i)14-s + (−0.0268 + 0.186i)15-s + (−0.654 + 0.755i)16-s + (−0.836 + 1.83i)17-s + ⋯ |
| L(s) = 1 | + (−0.594 − 0.382i)2-s + (−0.0821 − 0.571i)3-s + (0.207 + 0.454i)4-s + (−0.0808 − 0.0237i)5-s + (−0.169 + 0.371i)6-s + (−0.247 + 0.285i)7-s + (0.0503 − 0.349i)8-s + (−0.319 + 0.0939i)9-s + (0.0390 + 0.0450i)10-s + (1.14 − 0.737i)11-s + (0.242 − 0.156i)12-s + (0.220 + 0.253i)13-s + (0.256 − 0.0752i)14-s + (−0.00692 + 0.0481i)15-s + (−0.163 + 0.188i)16-s + (−0.202 + 0.444i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.387 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.953798 - 0.634029i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.953798 - 0.634029i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.841 + 0.540i)T \) |
| 3 | \( 1 + (0.142 + 0.989i)T \) |
| 7 | \( 1 + (0.654 - 0.755i)T \) |
| 23 | \( 1 + (-3.83 - 2.87i)T \) |
| good | 5 | \( 1 + (0.180 + 0.0530i)T + (4.20 + 2.70i)T^{2} \) |
| 11 | \( 1 + (-3.80 + 2.44i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-0.793 - 0.915i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.836 - 1.83i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-1.21 - 2.66i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-3.37 + 7.39i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.377 + 2.62i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-5.09 + 1.49i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-9.86 - 2.89i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.268 - 1.86i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 1.78T + 47T^{2} \) |
| 53 | \( 1 + (-4.22 + 4.87i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (2.81 + 3.24i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-1.02 + 7.10i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (11.6 + 7.46i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (5.25 + 3.37i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-1.15 - 2.53i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (2.11 + 2.44i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-15.9 + 4.69i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-0.116 - 0.807i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-9.60 - 2.81i)T + (81.6 + 52.4i)T^{2} \) |
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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
−9.670833487229040014427293260972, −9.119003676131227859473315637770, −8.211691303046701287967576482615, −7.55540497640559874573312330823, −6.33169011773801270014166472039, −5.98932242148100103822271283493, −4.30274160721674246157528600475, −3.32936144859313152798187960429, −2.08741866645052355073314597259, −0.850345535899231514287777035026,
1.09299214998843681259520641046, 2.78490052287160780077898796905, 4.04707956192677595044012135076, 4.91529998173502596812253962517, 6.01699565691248681426369655670, 6.93315079172601805014015781548, 7.51710975930220450763782509845, 8.917378848189203296459404939696, 9.120104084453094055531322980855, 10.09174889628478295460584462348
