L(s) = 1 | + (−1 − 1.73i)5-s + (1 − 1.73i)11-s + 4·13-s + (−3 + 5.19i)17-s + (4 + 6.92i)19-s + (−3 − 5.19i)23-s + (0.500 − 0.866i)25-s + 10·29-s + (2 − 3.46i)31-s + (−3 − 5.19i)37-s − 6·41-s + 4·43-s + (−4 − 6.92i)47-s + (1 − 1.73i)53-s − 3.99·55-s + ⋯ |
L(s) = 1 | + (−0.447 − 0.774i)5-s + (0.301 − 0.522i)11-s + 1.10·13-s + (−0.727 + 1.26i)17-s + (0.917 + 1.58i)19-s + (−0.625 − 1.08i)23-s + (0.100 − 0.173i)25-s + 1.85·29-s + (0.359 − 0.622i)31-s + (−0.493 − 0.854i)37-s − 0.937·41-s + 0.609·43-s + (−0.583 − 1.01i)47-s + (0.137 − 0.237i)53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.623397038\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.623397038\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4 - 6.92i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 31 | \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3 + 5.19i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1 + 1.73i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4 + 6.92i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 10T + 71T^{2} \) |
| 73 | \( 1 + (-2 + 3.46i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + (-7 - 12.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−8.929437560417049530649254143739, −8.279065044700320875082612490360, −8.055250166016411363199403881654, −6.52352173358036477463193674405, −6.12833013177810949718136754426, −5.03869372266193042569691126211, −4.06264070557333237350571060364, −3.49646891531374989788758422672, −1.92330405248765734572686974635, −0.75522123158939433778133858115,
1.11424805813388068093017046870, 2.67596701654423032832564116924, 3.35051073617788004427920111886, 4.47716268805571381575434138277, 5.23651417419348683409569018204, 6.55042227123339253103176996654, 6.90523684432299866215992131695, 7.71877403305596125776327766156, 8.701715136667653329038702155890, 9.360744082005070130876122216836