L(s) = 1 | + 5-s + (0.866 − 0.5i)7-s + i·11-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s + i·23-s + (−0.5 + 0.866i)29-s + (0.866 − 0.5i)35-s + (0.5 − 0.866i)37-s + (0.5 + 0.866i)41-s + (−0.866 − 0.5i)43-s + (0.499 − 0.866i)49-s + (−0.5 − 0.866i)53-s + i·55-s + ⋯ |
L(s) = 1 | + 5-s + (0.866 − 0.5i)7-s + i·11-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s + i·23-s + (−0.5 + 0.866i)29-s + (0.866 − 0.5i)35-s + (0.5 − 0.866i)37-s + (0.5 + 0.866i)41-s + (−0.866 − 0.5i)43-s + (0.499 − 0.866i)49-s + (−0.5 − 0.866i)53-s + i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.651038423\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.651038423\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.866 + 0.5i)T \) |
good | 5 | \( 1 - T + T^{2} \) |
| 11 | \( 1 - iT - T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 - iT - T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + 2iT - T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 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
−9.149866576654530072848035755148, −8.141501913736016458559848430347, −7.48688767735395361767893634131, −6.63816175228364708981597958813, −5.97362816935965176699709774405, −5.04427850440481505862219094649, −4.38336218551121490118980430582, −3.45945826878210643824807223350, −1.86144454536806154846744435504, −1.71885085858526978628349529123,
1.14135811186556960384391995694, 2.27283431008472217328534827076, 3.04024375250871161177008710990, 4.24274434810113407003037279543, 5.18187938329127415530572722742, 5.91378962935762865518366610226, 6.21746511288709485677171394781, 7.52161872611361802735544279355, 8.239924773873139799000644763337, 8.737840460267887155699788413145