| L(s) = 1 | + (0.501 − 1.93i)2-s + (−0.502 − 2.95i)3-s + (−3.49 − 1.94i)4-s + (4.60 + 1.94i)5-s + (−5.97 − 0.511i)6-s + (8.02 − 8.02i)7-s + (−5.51 + 5.79i)8-s + (−8.49 + 2.97i)9-s + (6.07 − 7.94i)10-s + (4.00 − 12.3i)11-s + (−3.98 + 11.3i)12-s + (−8.15 − 4.15i)13-s + (−11.5 − 19.5i)14-s + (3.44 − 14.5i)15-s + (8.45 + 13.5i)16-s + (2.48 − 15.7i)17-s + ⋯ |
| L(s) = 1 | + (0.250 − 0.968i)2-s + (−0.167 − 0.985i)3-s + (−0.874 − 0.485i)4-s + (0.921 + 0.389i)5-s + (−0.996 − 0.0852i)6-s + (1.14 − 1.14i)7-s + (−0.689 + 0.724i)8-s + (−0.943 + 0.330i)9-s + (0.607 − 0.794i)10-s + (0.363 − 1.11i)11-s + (−0.332 + 0.943i)12-s + (−0.627 − 0.319i)13-s + (−0.822 − 1.39i)14-s + (0.229 − 0.973i)15-s + (0.528 + 0.848i)16-s + (0.146 − 0.923i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0554i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0510242 - 1.83957i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0510242 - 1.83957i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.501 + 1.93i)T \) |
| 3 | \( 1 + (0.502 + 2.95i)T \) |
| 5 | \( 1 + (-4.60 - 1.94i)T \) |
| good | 7 | \( 1 + (-8.02 + 8.02i)T - 49iT^{2} \) |
| 11 | \( 1 + (-4.00 + 12.3i)T + (-97.8 - 71.1i)T^{2} \) |
| 13 | \( 1 + (8.15 + 4.15i)T + (99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (-2.48 + 15.7i)T + (-274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (14.2 - 10.3i)T + (111. - 343. i)T^{2} \) |
| 23 | \( 1 + (-16.0 - 31.5i)T + (-310. + 427. i)T^{2} \) |
| 29 | \( 1 + (6.70 + 4.87i)T + (259. + 799. i)T^{2} \) |
| 31 | \( 1 + (21.1 + 29.1i)T + (-296. + 913. i)T^{2} \) |
| 37 | \( 1 + (25.1 - 49.4i)T + (-804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-17.6 + 5.73i)T + (1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (-15.0 - 15.0i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (4.92 + 31.1i)T + (-2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (-3.63 - 22.9i)T + (-2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (-79.7 + 25.9i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (6.73 - 20.7i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (-15.7 + 99.6i)T + (-4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (-30.3 - 22.0i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (4.00 + 7.86i)T + (-3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (18.5 + 13.4i)T + (1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-15.5 + 98.4i)T + (-6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (3.09 - 9.52i)T + (-6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-4.07 - 25.7i)T + (-8.94e3 + 2.90e3i)T^{2} \) |
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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
−11.19066467238949260400431105307, −10.47409775301242713448726617199, −9.350269868983371348252211496040, −8.181240347369688140421288927379, −7.18363782131910679975014038058, −5.88273200420762115829160393457, −5.00323460582609439338753954745, −3.36648233055118327700827799418, −1.98388099732063261876436004162, −0.893615588276528815083243280710,
2.27412133086201123930080387975, 4.33312620431834566830356195635, 5.01207206855739629182222773699, 5.76263716419604065093637117575, 6.92518302075360086523577252132, 8.555286063722987262280928453064, 8.897044561066958560878437723568, 9.831530248644030090584151855627, 10.92664877021231337009565999037, 12.30525821044751766062322463923
