L(s) = 1 | + (−1.5 − 2.59i)5-s + (0.5 − 0.866i)7-s + (−1.5 + 2.59i)11-s + (−0.5 − 0.866i)13-s − 6·17-s + 4·19-s + (−1.5 − 2.59i)23-s + (−2 + 3.46i)25-s + (−1.5 + 2.59i)29-s + (−2.5 − 4.33i)31-s − 3·35-s − 2·37-s + (1.5 + 2.59i)41-s + (−0.5 + 0.866i)43-s + (−4.5 + 7.79i)47-s + ⋯ |
L(s) = 1 | + (−0.670 − 1.16i)5-s + (0.188 − 0.327i)7-s + (−0.452 + 0.783i)11-s + (−0.138 − 0.240i)13-s − 1.45·17-s + 0.917·19-s + (−0.312 − 0.541i)23-s + (−0.400 + 0.692i)25-s + (−0.278 + 0.482i)29-s + (−0.449 − 0.777i)31-s − 0.507·35-s − 0.328·37-s + (0.234 + 0.405i)41-s + (−0.0762 + 0.132i)43-s + (−0.656 + 1.13i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.5 - 7.79i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.5 - 11.2i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.5 + 6.06i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.5 + 7.79i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + (5.5 - 9.52i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.896638656111439619872880877728, −7.932506839450750136561253281174, −7.51949580913777253593583752806, −6.49133796553145479714456176018, −5.33524603553496783036987217920, −4.60904772100616062882817729302, −4.07185540848452660517665422806, −2.67959578799257357047500303303, −1.38629320182771795568895917193, 0,
2.00957822122610717881840022745, 3.06075514713325963722460260782, 3.75082639121020168292577485544, 4.91356896588412488798167838108, 5.83269159480779106557759574530, 6.77139654011188818892121184951, 7.33797396553104296452270534206, 8.196861099471919525723328454638, 8.916202765149935336340970175563