L(s) = 1 | + (−1.58 + 1.22i)2-s + (1.00 − 3.87i)4-s + (3.70 − 6.41i)5-s + (8.20 − 4.73i)7-s + (3.16 + 7.34i)8-s + (1.99 + 14.6i)10-s + (−5.86 + 3.38i)11-s + (7.20 − 12.4i)13-s + (−7.17 + 17.5i)14-s + (−13.9 − 7.74i)16-s − 17.8·17-s − 5.19i·19-s + (−21.1 − 20.7i)20-s + (5.12 − 12.5i)22-s + (−21.5 − 12.4i)23-s + ⋯ |
L(s) = 1 | + (−0.790 + 0.612i)2-s + (0.250 − 0.968i)4-s + (0.740 − 1.28i)5-s + (1.17 − 0.677i)7-s + (0.395 + 0.918i)8-s + (0.199 + 1.46i)10-s + (−0.533 + 0.307i)11-s + (0.554 − 0.960i)13-s + (−0.512 + 1.25i)14-s + (−0.874 − 0.484i)16-s − 1.05·17-s − 0.273i·19-s + (−1.05 − 1.03i)20-s + (0.232 − 0.569i)22-s + (−0.938 − 0.542i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.430 + 0.902i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.430 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.09041 - 0.687700i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.09041 - 0.687700i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.58 - 1.22i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-3.70 + 6.41i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-8.20 + 4.73i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (5.86 - 3.38i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-7.20 + 12.4i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 17.8T + 289T^{2} \) |
| 19 | \( 1 + 5.19iT - 361T^{2} \) |
| 23 | \( 1 + (21.5 + 12.4i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-14.8 - 25.6i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-14.8 - 8.56i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 6.41T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-4.32 + 7.48i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (43.4 - 25.0i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-45.0 + 26.0i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 82.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + (72.5 + 41.8i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-4.20 - 7.28i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (25.1 + 14.5i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 113. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 2.16T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-129. + 74.7i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-11.7 + 6.77i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 17.8T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-69.1 - 119. i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−10.79434064652751151274495267746, −10.31416073732703395275245436265, −9.103115286067935575178625970768, −8.397811027729012147446806496863, −7.71055035342467366286140686421, −6.37687273487066191447605673188, −5.22028312728168379181737509531, −4.61875402377880230123497113787, −1.97776269187582703930780677367, −0.808448795261808097621852291926,
1.86168510156065146166439729596, 2.62138137609155731920359415889, 4.19011241054358338100753107429, 5.87144585628187871805130790929, 6.84914707959988858146701729394, 7.989983483638015274311483804132, 8.798878185632419020169379175658, 9.844440941276803575540594719677, 10.69349466448934170154800269952, 11.36375333687705017682029927448