L(s) = 1 | + (−0.5 − 0.866i)3-s − i·5-s + (0.866 + 0.5i)7-s + (−0.499 + 0.866i)9-s + 11-s + (0.866 + 0.5i)13-s + (−0.866 + 0.5i)15-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s − 0.999i·21-s + i·23-s + 0.999·27-s + (−0.866 + 0.5i)29-s + (−0.5 − 0.866i)33-s + (0.5 − 0.866i)35-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)3-s − i·5-s + (0.866 + 0.5i)7-s + (−0.499 + 0.866i)9-s + 11-s + (0.866 + 0.5i)13-s + (−0.866 + 0.5i)15-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s − 0.999i·21-s + i·23-s + 0.999·27-s + (−0.866 + 0.5i)29-s + (−0.5 − 0.866i)33-s + (0.5 − 0.866i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.194875650\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.194875650\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
good | 5 | \( 1 + iT - T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 - iT - T^{2} \) |
| 29 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 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
−8.888943587485255001387938938330, −8.589198797342687419657004015702, −7.61242454489178719072553868603, −6.97672274278337315943678076104, −5.83076104638045905265364285067, −5.41969378321553787935723681215, −4.54313980455455922116515121411, −3.39864777543165075511724832125, −1.74203795383929641651888265714, −1.29218108148143774963332762647,
1.28439242315768460532123388816, 2.92414905440633866336208172549, 3.78374022559574872057664383171, 4.45122840173151208539054881816, 5.48848765485006413602720253093, 6.29794826676785517414818559118, 6.92870572666423559025515893823, 7.955477714235190748248262014195, 8.730531829315067923124603784024, 9.587589548471870389599470092871