L(s) = 1 | + (1.12 + 0.650i)5-s + (−3.04 + 1.75i)7-s + (0.909 + 1.57i)11-s + (2.62 − 4.54i)13-s − 3.03i·17-s + 3.05i·19-s + (2.61 − 4.52i)23-s + (−1.65 − 2.86i)25-s + (3.24 − 1.87i)29-s + (3.77 + 2.17i)31-s − 4.57·35-s + 1.48·37-s + (−7.32 − 4.22i)41-s + (5.57 − 3.22i)43-s + (6.34 + 10.9i)47-s + ⋯ |
L(s) = 1 | + (0.504 + 0.291i)5-s + (−1.15 + 0.664i)7-s + (0.274 + 0.475i)11-s + (0.727 − 1.26i)13-s − 0.737i·17-s + 0.701i·19-s + (0.544 − 0.942i)23-s + (−0.330 − 0.572i)25-s + (0.603 − 0.348i)29-s + (0.677 + 0.391i)31-s − 0.773·35-s + 0.244·37-s + (−1.14 − 0.660i)41-s + (0.850 − 0.491i)43-s + (0.926 + 1.60i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.798617203\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.798617203\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.12 - 0.650i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (3.04 - 1.75i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.909 - 1.57i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.62 + 4.54i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 3.03iT - 17T^{2} \) |
| 19 | \( 1 - 3.05iT - 19T^{2} \) |
| 23 | \( 1 + (-2.61 + 4.52i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.24 + 1.87i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.77 - 2.17i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 1.48T + 37T^{2} \) |
| 41 | \( 1 + (7.32 + 4.22i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.57 + 3.22i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-6.34 - 10.9i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 11.2iT - 53T^{2} \) |
| 59 | \( 1 + (-0.342 + 0.594i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.07 - 3.60i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-10.9 - 6.33i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2.50T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 + (-1.80 + 1.04i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (8.58 + 14.8i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 12.5iT - 89T^{2} \) |
| 97 | \( 1 + (-4.39 - 7.61i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.961952422698758931627770078031, −8.243629113665205377822789289171, −7.30011577778479531330385812424, −6.36664843605992631801628796509, −6.02993832178097099357164161177, −5.13339869771533910199079696176, −4.00650055437392650224939722922, −2.97719996484185799407421508315, −2.43897014659484611305611413660, −0.841149957307762588212525254426,
0.869826517058519335205367793583, 1.98884045296079542520176031059, 3.35515367127351772838854613881, 3.86882635933181551751126389571, 4.94168787912381993685198008620, 5.92633855568762926024608640788, 6.60898085871308893570841259673, 7.05346700410772507530059799553, 8.271872883771580343816667529017, 8.942467222143808265592404375380