L(s) = 1 | + (−0.643 − 1.11i)2-s + (−0.5 + 0.866i)3-s + (0.171 − 0.296i)4-s + (−1.95 − 3.39i)5-s + 1.28·6-s + (−0.234 + 2.63i)7-s − 3.01·8-s + (−0.499 − 0.866i)9-s + (−2.52 + 4.36i)10-s + (0.5 − 0.866i)11-s + (0.171 + 0.296i)12-s − 3.04·13-s + (3.08 − 1.43i)14-s + 3.91·15-s + (1.59 + 2.76i)16-s + (−1.98 + 3.44i)17-s + ⋯ |
L(s) = 1 | + (−0.455 − 0.788i)2-s + (−0.288 + 0.499i)3-s + (0.0856 − 0.148i)4-s + (−0.875 − 1.51i)5-s + 0.525·6-s + (−0.0885 + 0.996i)7-s − 1.06·8-s + (−0.166 − 0.288i)9-s + (−0.797 + 1.38i)10-s + (0.150 − 0.261i)11-s + (0.0494 + 0.0856i)12-s − 0.843·13-s + (0.825 − 0.383i)14-s + 1.01·15-s + (0.399 + 0.692i)16-s + (−0.481 + 0.834i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.151i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 - 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0307296 + 0.403537i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0307296 + 0.403537i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.234 - 2.63i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
good | 2 | \( 1 + (0.643 + 1.11i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (1.95 + 3.39i)T + (-2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + 3.04T + 13T^{2} \) |
| 17 | \( 1 + (1.98 - 3.44i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.79 + 6.57i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.25 + 3.90i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 3.75T + 29T^{2} \) |
| 31 | \( 1 + (-3.37 + 5.83i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.171 + 0.296i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 2.79T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 + (0.828 + 1.43i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.47 + 11.2i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.83 + 3.17i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.234 - 0.405i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.28 + 2.23i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 5.00T + 71T^{2} \) |
| 73 | \( 1 + (-4.36 + 7.56i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.359 + 0.623i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 + (-6.17 - 10.6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−11.72294593244632598817469501305, −10.86847088658745242933774646122, −9.677783754543208706215787909273, −8.877639392898293611578766445871, −8.337783744399611859696992072970, −6.39266895174126317941057705265, −5.19687492371404517808090140627, −4.21052771748260298562045961778, −2.39691101137453934269770924319, −0.37462168265862799602655610699,
2.80079855704437018519362357019, 4.09395285296385722055944087953, 6.12406447630171194011383237585, 7.09198929617339378087978529030, 7.34952197952317701573599446982, 8.281146135391031738933654017527, 9.921774787860672459287891780376, 10.77522616575810057668168663455, 11.77763405584022896184656004037, 12.40795757076901436179449544623