L(s) = 1 | + (1.36 + 2.36i)2-s + (−2.71 + 4.69i)4-s + (1.09 + 1.89i)5-s + (−2.19 − 1.47i)7-s − 9.33·8-s + (−2.98 + 5.16i)10-s + (−0.524 + 0.907i)11-s + 13-s + (0.484 − 7.19i)14-s + (−7.29 − 12.6i)16-s + (−2.64 + 4.58i)17-s + (−0.378 − 0.655i)19-s − 11.8·20-s − 2.85·22-s + (0.326 + 0.566i)23-s + ⋯ |
L(s) = 1 | + (0.963 + 1.66i)2-s + (−1.35 + 2.34i)4-s + (0.489 + 0.847i)5-s + (−0.830 − 0.557i)7-s − 3.30·8-s + (−0.942 + 1.63i)10-s + (−0.158 + 0.273i)11-s + 0.277·13-s + (0.129 − 1.92i)14-s + (−1.82 − 3.16i)16-s + (−0.641 + 1.11i)17-s + (−0.0868 − 0.150i)19-s − 2.65·20-s − 0.609·22-s + (0.0681 + 0.118i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.796320 - 1.61411i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.796320 - 1.61411i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.19 + 1.47i)T \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + (-1.36 - 2.36i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (-1.09 - 1.89i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.524 - 0.907i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (2.64 - 4.58i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.378 + 0.655i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.326 - 0.566i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 3.10T + 29T^{2} \) |
| 31 | \( 1 + (0.513 - 0.890i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.44 - 9.43i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 7.32T + 41T^{2} \) |
| 43 | \( 1 - 0.887T + 43T^{2} \) |
| 47 | \( 1 + (-1.16 - 2.02i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.44 + 4.23i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.524 - 0.907i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.24 - 10.8i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.23 - 3.87i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6.60T + 71T^{2} \) |
| 73 | \( 1 + (-4.14 + 7.17i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.07 + 1.85i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6.66T + 83T^{2} \) |
| 89 | \( 1 + (2.88 + 4.99i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 2.88T + 97T^{2} \) |
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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
−10.63489592376342458641368860436, −9.822447322121454847047727481354, −8.740023986139112994046630652374, −7.938761422520236553183409769852, −6.85940094460222465675404456790, −6.57080815653259832699897288808, −5.82078031640372348578465421500, −4.64339041382421102423861127760, −3.74985029461224987741588712220, −2.79272861244120994395449572551,
0.63687257994219050547929030194, 2.07806099317137135277096499044, 2.96933909354590169025950693890, 4.03448045348364326218715405269, 5.07988887735436483834803221518, 5.64700126406870887425270298295, 6.63894781148425697860443756386, 8.544044587708975612607949612642, 9.323468891936888298493607912214, 9.696467483608795906668149052832