L(s) = 1 | + 2-s + (0.5 − 0.866i)3-s + 4-s + (0.5 − 0.866i)6-s + 8-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)12-s + 16-s + (−1.5 + 0.866i)17-s + (−0.499 − 0.866i)18-s − 19-s + (0.5 − 0.866i)24-s + (0.5 + 0.866i)25-s − 0.999·27-s + 32-s + ⋯ |
L(s) = 1 | + 2-s + (0.5 − 0.866i)3-s + 4-s + (0.5 − 0.866i)6-s + 8-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)12-s + 16-s + (−1.5 + 0.866i)17-s + (−0.499 − 0.866i)18-s − 19-s + (0.5 − 0.866i)24-s + (0.5 + 0.866i)25-s − 0.999·27-s + 32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.247133729\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.247133729\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 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 | \( 1 + 1.73iT - T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.645283325981158141300416177522, −8.603574794593597963786900524688, −8.022572272044741461334092658294, −6.89272122826084991012093978262, −6.56649810600448946036840716606, −5.61742885590298475158585292665, −4.47637269167217473955573615766, −3.60316751573377103670795345528, −2.52308499792742952572811717848, −1.67247916555918633457811860114,
2.19004075749737114168407515878, 2.90370896257695148726069880142, 4.13935682905790601537629219185, 4.54941426375695327588876820909, 5.51862397367057283459929811540, 6.49605851361768181737660137189, 7.31059235504777561464214283150, 8.375799794996604349316021392341, 9.035382161108632739797915950459, 10.05738317743343404393516529408