| L(s) = 1 | + (−0.252 − 0.909i)2-s + (−1.39 + 1.02i)3-s + (0.949 − 0.571i)4-s + (−2.96 + 0.160i)5-s + (1.28 + 1.01i)6-s + (−0.330 − 2.01i)7-s + (−2.13 − 2.01i)8-s + (0.893 − 2.86i)9-s + (0.894 + 2.65i)10-s + (−0.839 − 1.58i)11-s + (−0.738 + 1.77i)12-s + (−5.22 − 2.08i)13-s + (−1.75 + 0.810i)14-s + (3.96 − 3.26i)15-s + (−0.260 + 0.490i)16-s + (5.41 + 0.887i)17-s + ⋯ |
| L(s) = 1 | + (−0.178 − 0.643i)2-s + (−0.805 + 0.592i)3-s + (0.474 − 0.285i)4-s + (−1.32 + 0.0718i)5-s + (0.525 + 0.412i)6-s + (−0.124 − 0.762i)7-s + (−0.753 − 0.713i)8-s + (0.297 − 0.954i)9-s + (0.282 + 0.839i)10-s + (−0.253 − 0.477i)11-s + (−0.213 + 0.511i)12-s + (−1.44 − 0.577i)13-s + (−0.467 + 0.216i)14-s + (1.02 − 0.843i)15-s + (−0.0650 + 0.122i)16-s + (1.31 + 0.215i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.498i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.120194 - 0.450310i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.120194 - 0.450310i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.39 - 1.02i)T \) |
| 59 | \( 1 + (2.66 + 7.20i)T \) |
| good | 2 | \( 1 + (0.252 + 0.909i)T + (-1.71 + 1.03i)T^{2} \) |
| 5 | \( 1 + (2.96 - 0.160i)T + (4.97 - 0.540i)T^{2} \) |
| 7 | \( 1 + (0.330 + 2.01i)T + (-6.63 + 2.23i)T^{2} \) |
| 11 | \( 1 + (0.839 + 1.58i)T + (-6.17 + 9.10i)T^{2} \) |
| 13 | \( 1 + (5.22 + 2.08i)T + (9.43 + 8.94i)T^{2} \) |
| 17 | \( 1 + (-5.41 - 0.887i)T + (16.1 + 5.42i)T^{2} \) |
| 19 | \( 1 + (2.57 - 3.02i)T + (-3.07 - 18.7i)T^{2} \) |
| 23 | \( 1 + (2.77 + 2.11i)T + (6.15 + 22.1i)T^{2} \) |
| 29 | \( 1 + (-3.39 - 0.942i)T + (24.8 + 14.9i)T^{2} \) |
| 31 | \( 1 + (-4.87 + 4.14i)T + (5.01 - 30.5i)T^{2} \) |
| 37 | \( 1 + (0.945 + 0.998i)T + (-2.00 + 36.9i)T^{2} \) |
| 41 | \( 1 + (-2.99 - 3.93i)T + (-10.9 + 39.5i)T^{2} \) |
| 43 | \( 1 + (-0.467 - 0.247i)T + (24.1 + 35.5i)T^{2} \) |
| 47 | \( 1 + (0.0740 - 1.36i)T + (-46.7 - 5.08i)T^{2} \) |
| 53 | \( 1 + (-3.79 + 11.2i)T + (-42.1 - 32.0i)T^{2} \) |
| 61 | \( 1 + (-8.81 + 2.44i)T + (52.2 - 31.4i)T^{2} \) |
| 67 | \( 1 + (9.36 - 9.88i)T + (-3.62 - 66.9i)T^{2} \) |
| 71 | \( 1 + (8.56 + 0.464i)T + (70.5 + 7.67i)T^{2} \) |
| 73 | \( 1 + (3.71 + 8.03i)T + (-47.2 + 55.6i)T^{2} \) |
| 79 | \( 1 + (3.40 + 5.02i)T + (-29.2 + 73.3i)T^{2} \) |
| 83 | \( 1 + (9.01 - 1.98i)T + (75.3 - 34.8i)T^{2} \) |
| 89 | \( 1 + (3.95 - 14.2i)T + (-76.2 - 45.8i)T^{2} \) |
| 97 | \( 1 + (-0.344 + 0.744i)T + (-62.7 - 73.9i)T^{2} \) |
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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
−12.04031126606780207649385756157, −11.30695413699260498441722195100, −10.25667653782871743486261961697, −9.993924498506783835874252622793, −8.086680800554458762297219622975, −7.07553936758720119421707973015, −5.79049059423730660837140945127, −4.30783745386010118911845819647, −3.20652578233132654637061604749, −0.47413986995915624827150782497,
2.60130635827291556886475747484, 4.65678354036875977660106452116, 5.87561910132585229706917332440, 7.19124603522917978291339774641, 7.54990738160990949423122808045, 8.661390579528729600885340988996, 10.25169379154777623804143961732, 11.67864831273592698569017584051, 11.99638140624782750334197181803, 12.55145742680644913634378192801
