| L(s) = 1 | + (−0.841 − 0.540i)3-s + (−0.355 − 2.46i)5-s + (1.47 + 0.141i)7-s + (0.415 + 0.909i)9-s + (−5.48 − 2.19i)11-s + (−2.97 − 2.84i)13-s + (−1.03 + 2.26i)15-s + (−2.20 + 6.36i)17-s + (3.49 − 0.333i)19-s + (−1.16 − 0.918i)21-s + (0.152 − 3.20i)23-s + (−1.17 + 0.345i)25-s + (0.142 − 0.989i)27-s + (−3.60 + 6.24i)29-s + (1.09 − 1.04i)31-s + ⋯ |
| L(s) = 1 | + (−0.485 − 0.312i)3-s + (−0.158 − 1.10i)5-s + (0.559 + 0.0533i)7-s + (0.138 + 0.303i)9-s + (−1.65 − 0.662i)11-s + (−0.826 − 0.787i)13-s + (−0.267 + 0.586i)15-s + (−0.534 + 1.54i)17-s + (0.802 − 0.0766i)19-s + (−0.254 − 0.200i)21-s + (0.0318 − 0.668i)23-s + (−0.235 + 0.0691i)25-s + (0.0273 − 0.190i)27-s + (−0.669 + 1.15i)29-s + (0.195 − 0.186i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.144i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0302336 + 0.416123i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0302336 + 0.416123i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.841 + 0.540i)T \) |
| 67 | \( 1 + (6.24 - 5.28i)T \) |
| good | 5 | \( 1 + (0.355 + 2.46i)T + (-4.79 + 1.40i)T^{2} \) |
| 7 | \( 1 + (-1.47 - 0.141i)T + (6.87 + 1.32i)T^{2} \) |
| 11 | \( 1 + (5.48 + 2.19i)T + (7.96 + 7.59i)T^{2} \) |
| 13 | \( 1 + (2.97 + 2.84i)T + (0.618 + 12.9i)T^{2} \) |
| 17 | \( 1 + (2.20 - 6.36i)T + (-13.3 - 10.5i)T^{2} \) |
| 19 | \( 1 + (-3.49 + 0.333i)T + (18.6 - 3.59i)T^{2} \) |
| 23 | \( 1 + (-0.152 + 3.20i)T + (-22.8 - 2.18i)T^{2} \) |
| 29 | \( 1 + (3.60 - 6.24i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.09 + 1.04i)T + (1.47 - 30.9i)T^{2} \) |
| 37 | \( 1 + (-3.11 - 5.40i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.79 - 0.924i)T + (38.0 - 15.2i)T^{2} \) |
| 43 | \( 1 + (8.41 + 9.70i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + (9.68 - 4.99i)T + (27.2 - 38.2i)T^{2} \) |
| 53 | \( 1 + (-5.19 + 5.98i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (11.5 + 3.38i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-0.979 + 0.392i)T + (44.1 - 42.0i)T^{2} \) |
| 71 | \( 1 + (-0.204 - 0.589i)T + (-55.8 + 43.8i)T^{2} \) |
| 73 | \( 1 + (6.65 - 2.66i)T + (52.8 - 50.3i)T^{2} \) |
| 79 | \( 1 + (0.340 - 1.40i)T + (-70.2 - 36.1i)T^{2} \) |
| 83 | \( 1 + (-12.0 + 9.48i)T + (19.5 - 80.6i)T^{2} \) |
| 89 | \( 1 + (-14.0 + 9.02i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-2.46 - 4.27i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.04066649524405879261558657770, −8.605866428341447638953747038835, −8.228965533526943398019921415738, −7.42858586863031715619010693145, −6.11878767835774678067256113610, −5.10315566815894216246549327393, −4.86105423528202121201838078188, −3.17859655704116021871756600870, −1.71471114754901547367850465489, −0.20872108664316889875672285855,
2.20599328826532508893316894424, 3.16456571257798686733096462103, 4.69779749754988123245808243573, 5.13555342986468315180225083148, 6.45168280029308873973728531186, 7.43386768935589210818683063102, 7.70564067128944302197675522541, 9.340165693883216986716393636814, 9.926699665228629824453584136533, 10.73164837028847353891683870312
