L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s − 0.999·8-s + 11-s + (−0.5 + 0.866i)16-s + (1 − 1.73i)17-s + (−0.5 − 0.866i)19-s + (0.5 − 0.866i)22-s + (−0.5 − 0.866i)25-s + (0.499 + 0.866i)32-s + (−0.999 − 1.73i)34-s − 0.999·38-s + (−0.5 + 0.866i)41-s + (−1 + 1.73i)43-s + (−0.499 − 0.866i)44-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s − 0.999·8-s + 11-s + (−0.5 + 0.866i)16-s + (1 − 1.73i)17-s + (−0.5 − 0.866i)19-s + (0.5 − 0.866i)22-s + (−0.5 − 0.866i)25-s + (0.499 + 0.866i)32-s + (−0.999 − 1.73i)34-s − 0.999·38-s + (−0.5 + 0.866i)41-s + (−1 + 1.73i)43-s + (−0.499 − 0.866i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.323359642\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.323359642\) |
\(L(1)\) |
|
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 | \( 1 \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−9.592029729301271249953825001604, −9.103891131186112088069976664638, −8.053606816114263555851504370460, −6.90020675466280367805861618174, −6.15383790416726603854635110445, −5.09273343030216437140872411861, −4.40307498989974494847252125232, −3.34667348171865755274524163341, −2.45711534118218481231559164559, −1.05614377543293206627987499296,
1.80031926845812421099671004847, 3.62509479241877635208500933010, 3.85134957783059151022850704898, 5.20185141778349170490553110825, 5.95224180534742166519737696206, 6.62532180372079981880622406667, 7.55921138106756165977651633892, 8.310543646827297557422856591370, 8.965372255574945273045409780896, 9.909641356401712306348470799423