L(s) = 1 | + (−1 + 1.73i)5-s − 3·7-s + 6·11-s + (0.5 + 0.866i)13-s + (−1 + 1.73i)17-s + (4 − 1.73i)19-s + (0.500 + 0.866i)25-s + (−1 − 1.73i)29-s + 31-s + (3 − 5.19i)35-s − 7·37-s + (−0.5 + 0.866i)43-s + 2·49-s + (−2 − 3.46i)53-s + (−6 + 10.3i)55-s + ⋯ |
L(s) = 1 | + (−0.447 + 0.774i)5-s − 1.13·7-s + 1.80·11-s + (0.138 + 0.240i)13-s + (−0.242 + 0.420i)17-s + (0.917 − 0.397i)19-s + (0.100 + 0.173i)25-s + (−0.185 − 0.321i)29-s + 0.179·31-s + (0.507 − 0.878i)35-s − 1.15·37-s + (−0.0762 + 0.132i)43-s + 0.285·49-s + (−0.274 − 0.475i)53-s + (−0.809 + 1.40i)55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.325229045\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.325229045\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-4 + 1.73i)T \) |
good | 5 | \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2 + 3.46i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4 + 6.92i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.5 - 9.52i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.5 - 12.9i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (4.5 - 7.79i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.5 - 11.2i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 14T + 83T^{2} \) |
| 89 | \( 1 + (-6 - 10.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5 - 8.66i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.166364681294549300852532771614, −8.343902814544177251246791840366, −7.21891069205695543263464810000, −6.74235916654225749158340644987, −6.28225212430359841448048758840, −5.18886732773353441794442939639, −3.79026423806617108421397104815, −3.68330849469412559022062920112, −2.55189422188455684418291350211, −1.15352962241163900241119319056,
0.50175226794472828149829641733, 1.59536403484124578836069784750, 3.14966209925132612810778793289, 3.74982591028905006728197135366, 4.59947917396128513003517154896, 5.55342501522582746160853348767, 6.46676353902859107088307218351, 6.94953495799220625386005164377, 7.929373110075289372246179410398, 8.815751935529598399336026629237