L(s) = 1 | + (0.5 − 0.866i)3-s + 1.73i·7-s + (−0.499 − 0.866i)9-s + 3.46i·11-s + (−4.5 + 2.59i)13-s + (−3 + 5.19i)17-s + (4 + 1.73i)19-s + (1.49 + 0.866i)21-s + (3 − 1.73i)23-s + (2.5 + 4.33i)25-s − 0.999·27-s + (−3 + 1.73i)29-s − 31-s + (2.99 + 1.73i)33-s − 8.66i·37-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + 0.654i·7-s + (−0.166 − 0.288i)9-s + 1.04i·11-s + (−1.24 + 0.720i)13-s + (−0.727 + 1.26i)17-s + (0.917 + 0.397i)19-s + (0.327 + 0.188i)21-s + (0.625 − 0.361i)23-s + (0.5 + 0.866i)25-s − 0.192·27-s + (−0.557 + 0.321i)29-s − 0.179·31-s + (0.522 + 0.301i)33-s − 1.42i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.08491 + 0.791070i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08491 + 0.791070i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (-4 - 1.73i)T \) |
good | 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 - 1.73iT - 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 + (4.5 - 2.59i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-3 + 1.73i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3 - 1.73i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 + 8.66iT - 37T^{2} \) |
| 41 | \( 1 + (6 + 3.46i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.5 - 2.59i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-9 + 5.19i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (9 - 5.19i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.5 - 11.2i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.5 + 4.33i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 17.3iT - 83T^{2} \) |
| 89 | \( 1 + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-12 - 6.92i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.17797797541331261489630197096, −9.188384035569067849602779995837, −8.817726229646343670806965207565, −7.41491799265476924221784149956, −7.18239931262349631905017137322, −5.96489148589154992334314721336, −5.02020236308597844773629645785, −3.95399030046656422405723870926, −2.56706683003918643526761462452, −1.72615663917162609494456675295,
0.60431571564010818080565222292, 2.63445302399547831367277450341, 3.39197989490381672356821331362, 4.70725944386166004528507206543, 5.27046842458743859044182300000, 6.60533026588221298735142860939, 7.46676184608580131285050610550, 8.222510358578475099356258572971, 9.294022699367505094658789673751, 9.783271660277852020531979227690