L(s) = 1 | + (0.5 + 0.866i)3-s + 3.46i·5-s + (−0.499 + 0.866i)9-s + (3 − 1.73i)11-s + (−2.99 + 1.73i)15-s + (−3 + 5.19i)17-s + (6 + 3.46i)19-s − 6.99·25-s − 0.999·27-s + (3 + 5.19i)29-s − 6.92i·31-s + (3 + 1.73i)33-s + (−3 + 1.73i)41-s + (−4 + 6.92i)43-s + (−2.99 − 1.73i)45-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + 1.54i·5-s + (−0.166 + 0.288i)9-s + (0.904 − 0.522i)11-s + (−0.774 + 0.447i)15-s + (−0.727 + 1.26i)17-s + (1.37 + 0.794i)19-s − 1.39·25-s − 0.192·27-s + (0.557 + 0.964i)29-s − 1.24i·31-s + (0.522 + 0.301i)33-s + (−0.468 + 0.270i)41-s + (−0.609 + 1.05i)43-s + (−0.447 − 0.258i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.839305970\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.839305970\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 3.46iT - 5T^{2} \) |
| 7 | \( 1 + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-3 + 1.73i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-6 - 3.46i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 6.92iT - 31T^{2} \) |
| 37 | \( 1 + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3 - 1.73i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 3.46iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (-3 - 1.73i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-12 + 6.92i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (9 + 5.19i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 3.46iT - 83T^{2} \) |
| 89 | \( 1 + (15 - 8.66i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (6 + 3.46i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.602038550992661076315751589337, −8.643790016616586476999737706334, −7.914847969416035083546156294574, −7.00495377255021807437900498287, −6.33837141889202917714198143651, −5.63470144744477850661547212691, −4.29655760467187222311119057275, −3.51617357471633605780166713285, −2.91462084977314852045465562146, −1.64563484074332281157701551945,
0.65819238717337882180162583298, 1.56999233676276247093577321590, 2.77729138989949692675231982921, 4.02637316496171208846808786269, 4.87146959013622744319345932628, 5.43768700332791825306451630361, 6.73515739759209634828971089797, 7.20611415446508379634579859758, 8.246231823613105419138136300369, 8.885913778338953833279468821574