| L(s) = 1 | + (−0.607 + 1.27i)2-s + (−0.290 − 0.956i)3-s + (−1.26 − 1.55i)4-s + (0.269 + 2.73i)5-s + (1.39 + 0.210i)6-s + (0.857 − 1.28i)7-s + (2.74 − 0.669i)8-s + (−0.831 + 0.555i)9-s + (−3.65 − 1.31i)10-s + (−5.50 + 2.94i)11-s + (−1.11 + 1.65i)12-s + (5.53 + 0.545i)13-s + (1.11 + 1.87i)14-s + (2.53 − 1.05i)15-s + (−0.813 + 3.91i)16-s + (3.96 + 1.64i)17-s + ⋯ |
| L(s) = 1 | + (−0.429 + 0.903i)2-s + (−0.167 − 0.552i)3-s + (−0.631 − 0.775i)4-s + (0.120 + 1.22i)5-s + (0.570 + 0.0859i)6-s + (0.324 − 0.484i)7-s + (0.971 − 0.236i)8-s + (−0.277 + 0.185i)9-s + (−1.15 − 0.415i)10-s + (−1.66 + 0.887i)11-s + (−0.322 + 0.478i)12-s + (1.53 + 0.151i)13-s + (0.298 + 0.500i)14-s + (0.654 − 0.271i)15-s + (−0.203 + 0.979i)16-s + (0.961 + 0.398i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.458 - 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.454708 + 0.746036i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.454708 + 0.746036i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.607 - 1.27i)T \) |
| 3 | \( 1 + (0.290 + 0.956i)T \) |
| good | 5 | \( 1 + (-0.269 - 2.73i)T + (-4.90 + 0.975i)T^{2} \) |
| 7 | \( 1 + (-0.857 + 1.28i)T + (-2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (5.50 - 2.94i)T + (6.11 - 9.14i)T^{2} \) |
| 13 | \( 1 + (-5.53 - 0.545i)T + (12.7 + 2.53i)T^{2} \) |
| 17 | \( 1 + (-3.96 - 1.64i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (2.47 - 2.03i)T + (3.70 - 18.6i)T^{2} \) |
| 23 | \( 1 + (1.24 - 6.25i)T + (-21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (1.87 - 3.50i)T + (-16.1 - 24.1i)T^{2} \) |
| 31 | \( 1 + (1.60 - 1.60i)T - 31iT^{2} \) |
| 37 | \( 1 + (3.90 - 4.75i)T + (-7.21 - 36.2i)T^{2} \) |
| 41 | \( 1 + (-1.19 - 0.237i)T + (37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (1.18 - 3.92i)T + (-35.7 - 23.8i)T^{2} \) |
| 47 | \( 1 + (-4.43 + 10.7i)T + (-33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (3.21 + 6.02i)T + (-29.4 + 44.0i)T^{2} \) |
| 59 | \( 1 + (-12.4 + 1.22i)T + (57.8 - 11.5i)T^{2} \) |
| 61 | \( 1 + (-0.672 + 0.203i)T + (50.7 - 33.8i)T^{2} \) |
| 67 | \( 1 + (-5.38 + 1.63i)T + (55.7 - 37.2i)T^{2} \) |
| 71 | \( 1 + (4.06 + 2.71i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (3.68 + 5.51i)T + (-27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (1.22 + 2.95i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-7.62 - 9.29i)T + (-16.1 + 81.4i)T^{2} \) |
| 89 | \( 1 + (1.91 + 9.62i)T + (-82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (-9.90 + 9.90i)T - 97iT^{2} \) |
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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
−11.29606705237601658572732812477, −10.49089427128219967819285510178, −10.06537955067865894083840891687, −8.499557134236903525741137503154, −7.68974954881809328393822416611, −7.09001064150637642140449778086, −6.10349192907397394295325836166, −5.21822306900494982633526619338, −3.57442055351003084509702513640, −1.72789281690152438924908043047,
0.72097811215426068956189880854, 2.54404031489407953285869496402, 3.90172335711287525591577043122, 5.06264526069937073583199824029, 5.76356295217140895999699570895, 7.916134738096626354134253732946, 8.558802387441665291239498651216, 9.093834754140731380916825748347, 10.34852669902983867129912564143, 10.88528712166814419964647435406
