L(s) = 1 | + (−1.5 + 2.59i)3-s + (1 − 1.73i)5-s + 2·7-s + (−3 − 5.19i)9-s − 3·11-s + (1 + 1.73i)13-s + (3 + 5.19i)15-s + (3 − 5.19i)17-s + (3.5 + 2.59i)19-s + (−3 + 5.19i)21-s + (−2 − 3.46i)23-s + (0.500 + 0.866i)25-s + 9·27-s + (−3 − 5.19i)29-s + 8·31-s + ⋯ |
L(s) = 1 | + (−0.866 + 1.49i)3-s + (0.447 − 0.774i)5-s + 0.755·7-s + (−1 − 1.73i)9-s − 0.904·11-s + (0.277 + 0.480i)13-s + (0.774 + 1.34i)15-s + (0.727 − 1.26i)17-s + (0.802 + 0.596i)19-s + (−0.654 + 1.13i)21-s + (−0.417 − 0.722i)23-s + (0.100 + 0.173i)25-s + 1.73·27-s + (−0.557 − 0.964i)29-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.382540414\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.382540414\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-3.5 - 2.59i)T \) |
good | 3 | \( 1 + (1.5 - 2.59i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4 + 6.92i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-5 - 8.66i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1 - 1.73i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.5 + 6.06i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1 - 1.73i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (5.5 - 9.52i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 15T + 83T^{2} \) |
| 89 | \( 1 + (-9 - 15.5i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.5 + 2.59i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.831357223942675822520507549263, −9.305751994251309825856240194762, −8.370900778444348654450894934352, −7.44951945779846656210046650170, −5.94174340944807833801977567585, −5.46984632442411190606259456080, −4.73169985869724791905039860160, −4.11322932448793540072501119738, −2.69558462052687484145779530331, −0.886290636852528753201321402388,
1.02148054308135995621870827752, 2.06976718773429540194391168478, 3.10484149135912992263983793789, 4.85545938855094899517780733150, 5.77479697604734160832670129259, 6.18783085175458604362242869517, 7.25644334482054619575657468013, 7.77685665908670027529401752847, 8.434182535856960275161287202316, 9.958842003803732604735158127500