L(s) = 1 | + (1.57 − 0.724i)3-s + (1.94 − 2.28i)9-s + (−0.476 − 0.275i)11-s − 2.36i·17-s + 5.97·19-s + (2.5 − 4.33i)25-s + (1.41 − 5.00i)27-s + (−0.949 − 0.0874i)33-s + (9.39 − 5.42i)41-s + (−2.20 + 3.82i)43-s + (−3.5 − 6.06i)49-s + (−1.71 − 3.72i)51-s + (9.39 − 4.33i)57-s + (−13.2 + 7.62i)59-s + (8.18 + 14.1i)67-s + ⋯ |
L(s) = 1 | + (0.908 − 0.418i)3-s + (0.649 − 0.760i)9-s + (−0.143 − 0.0829i)11-s − 0.574i·17-s + 1.37·19-s + (0.5 − 0.866i)25-s + (0.272 − 0.962i)27-s + (−0.165 − 0.0152i)33-s + (1.46 − 0.847i)41-s + (−0.336 + 0.583i)43-s + (−0.5 − 0.866i)49-s + (−0.240 − 0.521i)51-s + (1.24 − 0.573i)57-s + (−1.71 + 0.992i)59-s + (0.999 + 1.73i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.635 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.300123603\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.300123603\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.57 + 0.724i)T \) |
good | 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.476 + 0.275i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 2.36iT - 17T^{2} \) |
| 19 | \( 1 - 5.97T + 19T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + (-9.39 + 5.42i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.20 - 3.82i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + (13.2 - 7.62i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.18 - 14.1i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (15.5 + 9i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 5.65iT - 89T^{2} \) |
| 97 | \( 1 + (4.84 - 8.39i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.541771821618554955991375895789, −8.877874230336528661065872226874, −7.988494291147488627777172710412, −7.34867797583922500520659802644, −6.53178642651605443969693898930, −5.43228268516983278580398916886, −4.33286671966794267699294914555, −3.25269536384947346612010067438, −2.42448001308019327104253529481, −1.01823845388743465153627281204,
1.49957804264462971910686255714, 2.80931741383333954480511397684, 3.60905344942569948552336970690, 4.65703816589287967013136897060, 5.51413959007195103380117414036, 6.73109571147427992149091890624, 7.68243233113176325532141104354, 8.196114210230298954171068930626, 9.353818652291032384045690074543, 9.571494639610330251229880559892