L(s) = 1 | + (1.17 − 2.02i)5-s − 3.83·7-s + 3.34·11-s + (−3.08 − 5.34i)13-s + (2.59 − 4.50i)17-s + (3.01 + 3.14i)19-s + (−1.17 − 2.02i)23-s + (−0.244 − 0.423i)25-s + (0.0250 + 0.0434i)29-s + 3.43·31-s + (−4.48 + 7.77i)35-s − 5.43·37-s + (−3.64 + 6.31i)41-s + (−4.43 + 7.67i)43-s + (−5.36 − 9.29i)47-s + ⋯ |
L(s) = 1 | + (0.523 − 0.907i)5-s − 1.44·7-s + 1.00·11-s + (−0.856 − 1.48i)13-s + (0.630 − 1.09i)17-s + (0.691 + 0.722i)19-s + (−0.244 − 0.423i)23-s + (−0.0489 − 0.0847i)25-s + (0.00466 + 0.00807i)29-s + 0.617·31-s + (−0.758 + 1.31i)35-s − 0.894·37-s + (−0.569 + 0.986i)41-s + (−0.675 + 1.17i)43-s + (−0.783 − 1.35i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 + 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.899 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.017088864\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.017088864\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-3.01 - 3.14i)T \) |
good | 5 | \( 1 + (-1.17 + 2.02i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 3.83T + 7T^{2} \) |
| 11 | \( 1 - 3.34T + 11T^{2} \) |
| 13 | \( 1 + (3.08 + 5.34i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.59 + 4.50i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (1.17 + 2.02i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.0250 - 0.0434i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 3.43T + 31T^{2} \) |
| 37 | \( 1 + 5.43T + 37T^{2} \) |
| 41 | \( 1 + (3.64 - 6.31i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.43 - 7.67i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.36 + 9.29i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.59 + 2.76i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.92 - 3.33i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.46 + 2.53i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.75 + 9.97i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.744 - 1.28i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.84 + 6.65i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.0875 + 0.151i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 7.00T + 83T^{2} \) |
| 89 | \( 1 + (6.77 + 11.7i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.84 + 3.19i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.588275469673017508743184541301, −7.77935932562736346734754442912, −6.89548195701132229736672936521, −6.16568094804870054525835945256, −5.38348131893205184473243155297, −4.75615735146348148548041409403, −3.41099635813918224858547667214, −2.95696627051738520714635018284, −1.43826425333825599818937882611, −0.32750099656648494969544384122,
1.57506651254605296634772252973, 2.65291303451834435554026003408, 3.46456621445416204803558983669, 4.22536350273921261510173141294, 5.46957391688592352903961302833, 6.37312319338804193165201994167, 6.75031966525609089897460557730, 7.27575868127544790524871371259, 8.572665688297385851286314467314, 9.391590415620753487136940832924