| L(s) = 1 | + (−1.13 − 1.96i)2-s + (1.99 + 0.726i)3-s + (−1.56 + 2.71i)4-s + (0.184 − 0.155i)5-s + (−0.836 − 4.74i)6-s + (2.60 − 2.18i)7-s + 2.58·8-s + (1.16 + 0.974i)9-s + (−0.513 − 0.186i)10-s + (0.418 − 0.152i)11-s + (−5.11 + 4.28i)12-s + (−3.69 + 3.10i)13-s + (−7.23 − 2.63i)14-s + (0.481 − 0.175i)15-s + (0.210 + 0.365i)16-s + (−3.08 − 5.34i)17-s + ⋯ |
| L(s) = 1 | + (−0.801 − 1.38i)2-s + (1.15 + 0.419i)3-s + (−0.784 + 1.35i)4-s + (0.0826 − 0.0693i)5-s + (−0.341 − 1.93i)6-s + (0.983 − 0.824i)7-s + 0.913·8-s + (0.387 + 0.324i)9-s + (−0.162 − 0.0591i)10-s + (0.126 − 0.0459i)11-s + (−1.47 + 1.23i)12-s + (−1.02 + 0.860i)13-s + (−1.93 − 0.703i)14-s + (0.124 − 0.0452i)15-s + (0.0526 + 0.0912i)16-s + (−0.748 − 1.29i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.734615 - 0.646658i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.734615 - 0.646658i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 109 | \( 1 + (-3.77 + 9.73i)T \) |
| good | 2 | \( 1 + (1.13 + 1.96i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.99 - 0.726i)T + (2.29 + 1.92i)T^{2} \) |
| 5 | \( 1 + (-0.184 + 0.155i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-2.60 + 2.18i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.418 + 0.152i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (3.69 - 3.10i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (3.08 + 5.34i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.39 - 4.13i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.10 - 3.63i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.38 - 1.59i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (0.471 + 0.395i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-5.79 - 4.86i)T + (6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + 9.81T + 41T^{2} \) |
| 43 | \( 1 + (4.09 + 7.08i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.934 - 5.29i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-1.43 + 1.20i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-0.843 + 0.306i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (1.05 + 5.96i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (6.92 + 5.81i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (6.92 - 11.9i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-13.3 + 4.85i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (1.92 + 1.61i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-7.86 - 2.86i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (1.08 + 6.14i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-5.78 + 4.85i)T + (16.8 - 95.5i)T^{2} \) |
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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
−13.59044917658379669593167548591, −11.95380539551592299733080700728, −11.32905143469138773706746821153, −10.00119983440528893956877880958, −9.445640279644021571914886596295, −8.421544500158854893289602575155, −7.40084546768022718398615043136, −4.59940149749423830864534023288, −3.30957988359406092230712791709, −1.85058218463444043232031769901,
2.42679961907080475435226206517, 5.00811779195445380270892321853, 6.40066673142018435725173098108, 7.71112694499583072353471278727, 8.314344122301121289504261569187, 9.005085563487386129866779831359, 10.27607115927890487661054272697, 11.97208177971112616178194125229, 13.29738319051213414420420396682, 14.52056578723875116221221464420
