L(s) = 1 | + (0.866 − 1.5i)2-s + (−1 − 1.73i)4-s − 1.73·8-s + (−0.866 − 1.5i)11-s + (−0.5 + 0.866i)16-s − 3·22-s + (−0.5 − 0.866i)25-s + (−0.5 + 0.866i)37-s + 43-s + (−1.73 + 3i)44-s − 1.73·50-s + (−0.866 − 1.5i)53-s − 0.999·64-s + (0.5 + 0.866i)67-s − 1.73·71-s + ⋯ |
L(s) = 1 | + (0.866 − 1.5i)2-s + (−1 − 1.73i)4-s − 1.73·8-s + (−0.866 − 1.5i)11-s + (−0.5 + 0.866i)16-s − 3·22-s + (−0.5 − 0.866i)25-s + (−0.5 + 0.866i)37-s + 43-s + (−1.73 + 3i)44-s − 1.73·50-s + (−0.866 − 1.5i)53-s − 0.999·64-s + (0.5 + 0.866i)67-s − 1.73·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.541248885\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.541248885\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + 1.73T + T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 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
−8.398971281812037569437184215340, −7.61999557829390956379270184286, −6.36843082560268455770981563537, −5.71474397797089338983141582548, −5.03739806561117057335019659410, −4.22865586431004958468657100943, −3.35897459107896017395468529004, −2.81875103999286883267258364009, −1.87665003720305507883228671364, −0.62284213264159484691769096461,
1.93930073926760297653388806448, 3.09847668027719573965562988434, 4.11597796690524910150070249510, 4.69596103101586496822016335487, 5.41058071613123887342595419718, 6.01645291934499231668423535754, 6.93547353466642123682207585027, 7.48451470406503086417831299777, 7.84651258200414070771364160517, 8.836991074674382989909616313813