L(s) = 1 | + (0.550 − 1.30i)2-s + (−1.39 − 1.43i)4-s + (−0.162 − 2.23i)5-s + 2.93·7-s + (−2.63 + 1.02i)8-s + (−2.99 − 1.01i)10-s + (0.663 − 0.663i)11-s + (1.12 − 1.12i)13-s + (1.61 − 3.82i)14-s + (−0.112 + 3.99i)16-s − 7.47i·17-s + (0.423 + 0.423i)19-s + (−2.97 + 3.34i)20-s + (−0.499 − 1.23i)22-s − 6.17·23-s + ⋯ |
L(s) = 1 | + (0.389 − 0.921i)2-s + (−0.697 − 0.716i)4-s + (−0.0724 − 0.997i)5-s + 1.10·7-s + (−0.931 + 0.363i)8-s + (−0.946 − 0.321i)10-s + (0.200 − 0.200i)11-s + (0.312 − 0.312i)13-s + (0.431 − 1.02i)14-s + (−0.0281 + 0.999i)16-s − 1.81i·17-s + (0.0971 + 0.0971i)19-s + (−0.664 + 0.747i)20-s + (−0.106 − 0.262i)22-s − 1.28·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.906 + 0.423i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.906 + 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.372807 - 1.67877i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.372807 - 1.67877i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.550 + 1.30i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.162 + 2.23i)T \) |
good | 7 | \( 1 - 2.93T + 7T^{2} \) |
| 11 | \( 1 + (-0.663 + 0.663i)T - 11iT^{2} \) |
| 13 | \( 1 + (-1.12 + 1.12i)T - 13iT^{2} \) |
| 17 | \( 1 + 7.47iT - 17T^{2} \) |
| 19 | \( 1 + (-0.423 - 0.423i)T + 19iT^{2} \) |
| 23 | \( 1 + 6.17T + 23T^{2} \) |
| 29 | \( 1 + (-2.95 - 2.95i)T + 29iT^{2} \) |
| 31 | \( 1 - 1.82T + 31T^{2} \) |
| 37 | \( 1 + (5.53 + 5.53i)T + 37iT^{2} \) |
| 41 | \( 1 - 12.3iT - 41T^{2} \) |
| 43 | \( 1 + (-0.897 - 0.897i)T + 43iT^{2} \) |
| 47 | \( 1 - 4.12iT - 47T^{2} \) |
| 53 | \( 1 + (0.146 + 0.146i)T + 53iT^{2} \) |
| 59 | \( 1 + (-7.72 + 7.72i)T - 59iT^{2} \) |
| 61 | \( 1 + (7.37 + 7.37i)T + 61iT^{2} \) |
| 67 | \( 1 + (-8.68 + 8.68i)T - 67iT^{2} \) |
| 71 | \( 1 - 8.95iT - 71T^{2} \) |
| 73 | \( 1 - 0.174T + 73T^{2} \) |
| 79 | \( 1 - 3.06T + 79T^{2} \) |
| 83 | \( 1 + (-9.18 + 9.18i)T - 83iT^{2} \) |
| 89 | \( 1 + 8.71iT - 89T^{2} \) |
| 97 | \( 1 - 10.5iT - 97T^{2} \) |
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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
−10.06140718246108506957423596164, −9.302123168309388342921067518471, −8.478901895496292440948386361497, −7.73829099637066106504253889891, −6.15739896668122556839314750971, −5.05424527821575975716698106134, −4.67917875228106388580976152165, −3.43867329677396487738258603303, −2.01340711675796226887478849197, −0.832385091141064246515719668877,
2.04367988285871888897768744378, 3.65033088562178661603489614886, 4.33937781010138201415252589293, 5.61180313564368281691961056701, 6.36693922636923870551475095044, 7.22693902176541669221505561577, 8.130395247701249253269784234037, 8.626207530084492832139340026367, 9.996625326968417406438445184903, 10.71420026960731184458526000116