L(s) = 1 | + (0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.499 − 0.866i)6-s − 5·7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (0.499 + 0.866i)10-s + 11-s − 0.999·12-s + (−3 − 5.19i)13-s + (−2.5 + 4.33i)14-s + (0.499 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (−2 + 3.46i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.223 + 0.387i)5-s + (−0.204 − 0.353i)6-s − 1.88·7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.158 + 0.273i)10-s + 0.301·11-s − 0.288·12-s + (−0.832 − 1.44i)13-s + (−0.668 + 1.15i)14-s + (0.129 + 0.223i)15-s + (−0.125 + 0.216i)16-s + (−0.485 + 0.840i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 - 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.117037 + 0.541519i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.117037 + 0.541519i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (0.5 - 4.33i)T \) |
good | 7 | \( 1 + 5T + 7T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + (3 + 5.19i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (3.5 + 6.06i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 7T + 37T^{2} \) |
| 41 | \( 1 + (-2.5 + 4.33i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3 - 5.19i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (5.5 + 9.52i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4 + 6.92i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6 + 10.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1 - 1.73i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5 - 8.66i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 10T + 83T^{2} \) |
| 89 | \( 1 + (6.5 + 11.2i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.04197239434909792433143160226, −9.802336231105367834325175494434, −8.492503754776798021061565107724, −7.53415357675152026962172304638, −6.35151653539977994956861521099, −5.92652352799652711155296294435, −4.14101402507608497519415351861, −3.23698953885078968562930083930, −2.38477296262791958911702648945, −0.25043747681281072754802364254,
2.65775782371746890693081722965, 3.77351532917281946157175500922, 4.59545782583814197203488182503, 5.77800025558536022715943352408, 6.84407819521152970351372081234, 7.34614225563783647250492057108, 8.974280613717650388146540694003, 9.257169270914479051718540195140, 9.974155174672416300948973208773, 11.42869921966167518258910561516