| L(s) = 1 | + (−1.38 + 0.270i)2-s + (−1.71 + 0.220i)3-s + (1.85 − 0.749i)4-s + (−3.40 − 1.07i)5-s + (2.32 − 0.770i)6-s + (0.509 − 0.237i)7-s + (−2.37 + 1.54i)8-s + (2.90 − 0.757i)9-s + (5.01 + 0.570i)10-s + (1.18 + 0.619i)11-s + (−3.01 + 1.69i)12-s + (−2.56 + 4.03i)13-s + (−0.642 + 0.467i)14-s + (6.08 + 1.09i)15-s + (2.87 − 2.78i)16-s + (3.60 + 2.08i)17-s + ⋯ |
| L(s) = 1 | + (−0.981 + 0.190i)2-s + (−0.991 + 0.127i)3-s + (0.927 − 0.374i)4-s + (−1.52 − 0.480i)5-s + (0.949 − 0.314i)6-s + (0.192 − 0.0897i)7-s + (−0.838 + 0.545i)8-s + (0.967 − 0.252i)9-s + (1.58 + 0.180i)10-s + (0.358 + 0.186i)11-s + (−0.871 + 0.489i)12-s + (−0.712 + 1.11i)13-s + (−0.171 + 0.124i)14-s + (1.57 + 0.282i)15-s + (0.718 − 0.695i)16-s + (0.874 + 0.504i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.157 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.157 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.214094 - 0.182618i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.214094 - 0.182618i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.38 - 0.270i)T \) |
| 3 | \( 1 + (1.71 - 0.220i)T \) |
| good | 5 | \( 1 + (3.40 + 1.07i)T + (4.09 + 2.86i)T^{2} \) |
| 7 | \( 1 + (-0.509 + 0.237i)T + (4.49 - 5.36i)T^{2} \) |
| 11 | \( 1 + (-1.18 - 0.619i)T + (6.30 + 9.01i)T^{2} \) |
| 13 | \( 1 + (2.56 - 4.03i)T + (-5.49 - 11.7i)T^{2} \) |
| 17 | \( 1 + (-3.60 - 2.08i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.557 - 4.23i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (7.60 + 3.54i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (1.89 + 0.420i)T + (26.2 + 12.2i)T^{2} \) |
| 31 | \( 1 + (-5.45 + 1.98i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-1.06 + 8.08i)T + (-35.7 - 9.57i)T^{2} \) |
| 41 | \( 1 + (-2.21 + 1.55i)T + (14.0 - 38.5i)T^{2} \) |
| 43 | \( 1 + (1.04 + 0.545i)T + (24.6 + 35.2i)T^{2} \) |
| 47 | \( 1 + (-0.599 + 1.64i)T + (-36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-1.50 - 3.63i)T + (-37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (5.33 + 1.68i)T + (48.3 + 33.8i)T^{2} \) |
| 61 | \( 1 + (-9.37 + 10.2i)T + (-5.31 - 60.7i)T^{2} \) |
| 67 | \( 1 + (2.32 + 1.48i)T + (28.3 + 60.7i)T^{2} \) |
| 71 | \( 1 + (2.10 + 7.86i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (-3.33 + 12.4i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (9.83 - 1.73i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (0.0805 - 0.363i)T + (-75.2 - 35.0i)T^{2} \) |
| 89 | \( 1 + (-8.29 - 2.22i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (0.945 - 0.793i)T + (16.8 - 95.5i)T^{2} \) |
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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
−9.978416609806810851975930713712, −9.210157618083359752772541980288, −8.007669307699871045868940001657, −7.67804724647524348661943936994, −6.67046244109128455756901685816, −5.79554026861325958557954026812, −4.54874091139908044326725337678, −3.81635461463369619256043546201, −1.73480902556204436789633675536, −0.29375310868089385894891154521,
0.926793228787960311336159096597, 2.81020536436852299582885113661, 3.87039239567521510066918484841, 5.13141856944697778721666813600, 6.28166800137910052863590676397, 7.25859420052877807072340446770, 7.69619673785678215334377990601, 8.443939873102182049093083028772, 9.879397164507002940985520043744, 10.26982459214013800306229557044
