L(s) = 1 | + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)5-s + (−2 − 1.73i)7-s + (1 + 1.73i)9-s + (1.5 − 2.59i)11-s + 6·13-s + 0.999·15-s + (2.5 − 4.33i)17-s + (0.5 + 0.866i)19-s + (2.5 − 0.866i)21-s + (3.5 + 6.06i)23-s + (2 − 3.46i)25-s − 5·27-s − 2·29-s + (2.5 − 4.33i)31-s + ⋯ |
L(s) = 1 | + (−0.288 + 0.499i)3-s + (−0.223 − 0.387i)5-s + (−0.755 − 0.654i)7-s + (0.333 + 0.577i)9-s + (0.452 − 0.783i)11-s + 1.66·13-s + 0.258·15-s + (0.606 − 1.05i)17-s + (0.114 + 0.198i)19-s + (0.545 − 0.188i)21-s + (0.729 + 1.26i)23-s + (0.400 − 0.692i)25-s − 0.962·27-s − 0.371·29-s + (0.449 − 0.777i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.24894 - 0.159157i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.24894 - 0.159157i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
good | 3 | \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 + (-2.5 + 4.33i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.5 - 6.06i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.5 - 2.59i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (2.5 + 4.33i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-7.5 + 12.9i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.5 - 7.79i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (3.5 - 6.06i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + (3.5 + 6.06i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−11.13263748987796882547345526943, −10.15811908975369985644136033384, −9.394529252350323576506991990101, −8.397903179338401172075113885829, −7.39144969507486566533369515328, −6.28460647726241375220285160019, −5.32006504234613265303786687791, −4.10686815133672853469954152287, −3.29588785858334462259153216482, −1.02716428366983755191768007058,
1.36245434682377856455429550156, 3.11156906286050994004484301992, 4.12021583864442773724023589305, 5.78480789819972127917308638266, 6.48461306680478037712395659960, 7.18325214464915713201215110724, 8.524708730523330474502969610910, 9.248502463746238962950567047845, 10.32182053853069587911626101136, 11.17472346800257854414717563573