| L(s) = 1 | + (−0.919 − 1.07i)2-s + (1.25 + 1.87i)3-s + (−0.309 + 1.97i)4-s + (0.509 + 2.56i)5-s + (0.862 − 3.06i)6-s + (−1.78 − 4.31i)7-s + (2.40 − 1.48i)8-s + (−0.792 + 1.91i)9-s + (2.28 − 2.90i)10-s + (−0.337 − 0.225i)11-s + (−4.08 + 1.89i)12-s + (0.558 − 2.80i)13-s + (−2.99 + 5.88i)14-s + (−4.16 + 4.16i)15-s + (−3.80 − 1.22i)16-s + (2.50 + 2.50i)17-s + ⋯ |
| L(s) = 1 | + (−0.650 − 0.759i)2-s + (0.722 + 1.08i)3-s + (−0.154 + 0.987i)4-s + (0.228 + 1.14i)5-s + (0.351 − 1.25i)6-s + (−0.675 − 1.63i)7-s + (0.851 − 0.524i)8-s + (−0.264 + 0.637i)9-s + (0.722 − 0.918i)10-s + (−0.101 − 0.0679i)11-s + (−1.17 + 0.546i)12-s + (0.154 − 0.778i)13-s + (−0.799 + 1.57i)14-s + (−1.07 + 1.07i)15-s + (−0.951 − 0.306i)16-s + (0.607 + 0.607i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.782809 + 0.0866802i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.782809 + 0.0866802i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.919 + 1.07i)T \) |
| good | 3 | \( 1 + (-1.25 - 1.87i)T + (-1.14 + 2.77i)T^{2} \) |
| 5 | \( 1 + (-0.509 - 2.56i)T + (-4.61 + 1.91i)T^{2} \) |
| 7 | \( 1 + (1.78 + 4.31i)T + (-4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (0.337 + 0.225i)T + (4.20 + 10.1i)T^{2} \) |
| 13 | \( 1 + (-0.558 + 2.80i)T + (-12.0 - 4.97i)T^{2} \) |
| 17 | \( 1 + (-2.50 - 2.50i)T + 17iT^{2} \) |
| 19 | \( 1 + (2.54 + 0.506i)T + (17.5 + 7.27i)T^{2} \) |
| 23 | \( 1 + (2.78 + 1.15i)T + (16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (4.40 - 2.94i)T + (11.0 - 26.7i)T^{2} \) |
| 31 | \( 1 - 0.289iT - 31T^{2} \) |
| 37 | \( 1 + (-1.93 + 0.384i)T + (34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (5.97 + 2.47i)T + (28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (3.47 - 5.19i)T + (-16.4 - 39.7i)T^{2} \) |
| 47 | \( 1 + (-0.140 - 0.140i)T + 47iT^{2} \) |
| 53 | \( 1 + (-0.438 - 0.292i)T + (20.2 + 48.9i)T^{2} \) |
| 59 | \( 1 + (1.05 + 5.31i)T + (-54.5 + 22.5i)T^{2} \) |
| 61 | \( 1 + (-4.77 - 7.14i)T + (-23.3 + 56.3i)T^{2} \) |
| 67 | \( 1 + (-2.53 - 3.79i)T + (-25.6 + 61.8i)T^{2} \) |
| 71 | \( 1 + (-5.35 - 12.9i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-5.89 + 14.2i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (4.17 - 4.17i)T - 79iT^{2} \) |
| 83 | \( 1 + (-7.83 - 1.55i)T + (76.6 + 31.7i)T^{2} \) |
| 89 | \( 1 + (6.46 - 2.67i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + 16.2iT - 97T^{2} \) |
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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
−14.90951947846076217309001153680, −13.92368921238175848383876566116, −12.85431854057649673152525447899, −10.88237025832782793413989891083, −10.34507286150409297370477973164, −9.728680691988439321591552040087, −8.162625066668693067536506975415, −6.86759730664273071668926919853, −3.96052881029724691585998125441, −3.13888990244719961567125370948,
2.00280200121863803546582611119, 5.33588293410147693460016179644, 6.55049579093794733146849796793, 8.040172854184342791796177071719, 8.866735456824622024452154938626, 9.598796566554307646435698037630, 11.89983134196355030768853233974, 12.87379870985761544182466941938, 13.78956572482413340327137592073, 15.01570745655425118418065280662
