L(s) = 1 | + (0.766 − 0.642i)2-s + (0.939 + 0.342i)3-s + (0.173 − 0.984i)4-s + (0.939 − 0.342i)6-s + (0.5 − 0.866i)7-s + (−0.500 − 0.866i)8-s + (−1.53 − 1.28i)9-s + (3 + 5.19i)11-s + (0.499 − 0.866i)12-s + (4.69 − 1.71i)13-s + (−0.173 − 0.984i)14-s + (−0.939 − 0.342i)16-s + (2.29 − 1.92i)17-s − 2·18-s + (0.766 − 0.642i)21-s + (5.63 + 2.05i)22-s + ⋯ |
L(s) = 1 | + (0.541 − 0.454i)2-s + (0.542 + 0.197i)3-s + (0.0868 − 0.492i)4-s + (0.383 − 0.139i)6-s + (0.188 − 0.327i)7-s + (−0.176 − 0.306i)8-s + (−0.510 − 0.428i)9-s + (0.904 + 1.56i)11-s + (0.144 − 0.250i)12-s + (1.30 − 0.474i)13-s + (−0.0464 − 0.263i)14-s + (−0.234 − 0.0855i)16-s + (0.557 − 0.467i)17-s − 0.471·18-s + (0.167 − 0.140i)21-s + (1.20 + 0.437i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.35771 - 1.00570i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.35771 - 1.00570i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.766 + 0.642i)T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-0.939 - 0.342i)T + (2.29 + 1.92i)T^{2} \) |
| 5 | \( 1 + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.69 + 1.71i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.29 + 1.92i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.520 + 2.95i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (6.89 + 5.78i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.38 - 7.87i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.520 + 2.95i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (6.89 - 5.78i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (1.73 - 9.84i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (3.83 + 3.21i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.04 - 5.90i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-6.57 - 2.39i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (9.39 + 3.42i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-3 + 5.19i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (11.2 - 4.10i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-7.66 + 6.42i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.32333291336015939056876217026, −9.462223308103296050125455098668, −8.811764649754775872865697374151, −7.69700103541705228152731687514, −6.68413789410578776063457225762, −5.75737480159091690591088744348, −4.50349514610211168859969136946, −3.77875520861388078911615037336, −2.72730879810629324886427028574, −1.30911764998984673611734410859,
1.62761660157931603573099112697, 3.25043477833002597993343221795, 3.76368272517226243596758619668, 5.38940808284644924323253182810, 5.92919135477686423445285166829, 6.95248166317181133914816960102, 8.030956516790073268174276674711, 8.737708021451125369054613482876, 9.152139928531293859659410642776, 11.02030769698767977106289062594