| L(s) = 1 | + (−2.78 + 1.11i)3-s + (2.99 − 2.23i)5-s + (−3.97 + 2.61i)7-s + (6.53 − 6.18i)9-s + (−1.24 + 10.6i)11-s + (−1.49 − 5.00i)13-s + (−5.87 + 9.55i)15-s + (5.51 + 0.971i)17-s + (1.82 + 10.3i)19-s + (8.17 − 11.7i)21-s + (−8.74 + 13.2i)23-s + (−3.16 + 10.5i)25-s + (−11.3 + 24.5i)27-s + (−38.1 + 35.9i)29-s + (−0.509 + 8.73i)31-s + ⋯ |
| L(s) = 1 | + (−0.929 + 0.370i)3-s + (0.599 − 0.446i)5-s + (−0.567 + 0.373i)7-s + (0.726 − 0.687i)9-s + (−0.112 + 0.966i)11-s + (−0.115 − 0.384i)13-s + (−0.391 + 0.636i)15-s + (0.324 + 0.0571i)17-s + (0.0961 + 0.545i)19-s + (0.389 − 0.557i)21-s + (−0.380 + 0.577i)23-s + (−0.126 + 0.422i)25-s + (−0.420 + 0.907i)27-s + (−1.31 + 1.24i)29-s + (−0.0164 + 0.281i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.417 - 0.908i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.441630 + 0.688643i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.441630 + 0.688643i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.78 - 1.11i)T \) |
| good | 5 | \( 1 + (-2.99 + 2.23i)T + (7.17 - 23.9i)T^{2} \) |
| 7 | \( 1 + (3.97 - 2.61i)T + (19.4 - 44.9i)T^{2} \) |
| 11 | \( 1 + (1.24 - 10.6i)T + (-117. - 27.9i)T^{2} \) |
| 13 | \( 1 + (1.49 + 5.00i)T + (-141. + 92.8i)T^{2} \) |
| 17 | \( 1 + (-5.51 - 0.971i)T + (271. + 98.8i)T^{2} \) |
| 19 | \( 1 + (-1.82 - 10.3i)T + (-339. + 123. i)T^{2} \) |
| 23 | \( 1 + (8.74 - 13.2i)T + (-209. - 485. i)T^{2} \) |
| 29 | \( 1 + (38.1 - 35.9i)T + (48.8 - 839. i)T^{2} \) |
| 31 | \( 1 + (0.509 - 8.73i)T + (-954. - 111. i)T^{2} \) |
| 37 | \( 1 + (4.52 - 1.64i)T + (1.04e3 - 879. i)T^{2} \) |
| 41 | \( 1 + (3.69 - 15.6i)T + (-1.50e3 - 754. i)T^{2} \) |
| 43 | \( 1 + (-21.9 - 50.9i)T + (-1.26e3 + 1.34e3i)T^{2} \) |
| 47 | \( 1 + (52.3 - 3.04i)T + (2.19e3 - 256. i)T^{2} \) |
| 53 | \( 1 + (34.3 + 19.8i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (3.82 + 32.7i)T + (-3.38e3 + 802. i)T^{2} \) |
| 61 | \( 1 + (-85.7 - 43.0i)T + (2.22e3 + 2.98e3i)T^{2} \) |
| 67 | \( 1 + (-51.3 + 54.4i)T + (-261. - 4.48e3i)T^{2} \) |
| 71 | \( 1 + (-20.4 + 24.4i)T + (-875. - 4.96e3i)T^{2} \) |
| 73 | \( 1 + (-43.6 + 36.6i)T + (925. - 5.24e3i)T^{2} \) |
| 79 | \( 1 + (-31.4 + 7.44i)T + (5.57e3 - 2.80e3i)T^{2} \) |
| 83 | \( 1 + (-2.97 - 12.5i)T + (-6.15e3 + 3.09e3i)T^{2} \) |
| 89 | \( 1 + (-45.6 - 54.4i)T + (-1.37e3 + 7.80e3i)T^{2} \) |
| 97 | \( 1 + (-113. + 153. i)T + (-2.69e3 - 9.01e3i)T^{2} \) |
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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
−11.72572030275467533642949574579, −10.70309350408485146454837481139, −9.677055357743070660836094619698, −9.400852057119251287548904793950, −7.78914031341222110977115364939, −6.63741226906045757201760053269, −5.64080240777889099897730808342, −4.95248555154381527477592556472, −3.51483286215028952265791899031, −1.60495436769473810762276377239,
0.43299723368135703040898079711, 2.27138813791941301409667109813, 3.89561530825902679992894148302, 5.37896078966983485103380653127, 6.23023109904920770898438821472, 6.95037910789845811571094182590, 8.090919320406384861540635745403, 9.507692324769486837234690209091, 10.28404261569236365744887665611, 11.11165279951737352586061894133
