| L(s) = 1 | + (−0.807 − 0.565i)2-s + (−0.756 − 1.55i)3-s + (−0.351 − 0.966i)4-s + (−2.04 + 0.911i)5-s + (−0.270 + 1.68i)6-s + (−0.275 − 0.590i)7-s + (−0.772 + 2.88i)8-s + (−1.85 + 2.35i)9-s + (2.16 + 0.418i)10-s + (0.890 + 1.06i)11-s + (−1.24 + 1.27i)12-s + (−2.93 − 4.19i)13-s + (−0.111 + 0.632i)14-s + (2.96 + 2.49i)15-s + (0.677 − 0.568i)16-s + (−1.18 − 4.41i)17-s + ⋯ |
| L(s) = 1 | + (−0.570 − 0.399i)2-s + (−0.436 − 0.899i)3-s + (−0.175 − 0.483i)4-s + (−0.913 + 0.407i)5-s + (−0.110 + 0.688i)6-s + (−0.104 − 0.223i)7-s + (−0.273 + 1.01i)8-s + (−0.618 + 0.785i)9-s + (0.684 + 0.132i)10-s + (0.268 + 0.319i)11-s + (−0.357 + 0.369i)12-s + (−0.814 − 1.16i)13-s + (−0.0298 + 0.169i)14-s + (0.765 + 0.643i)15-s + (0.169 − 0.142i)16-s + (−0.286 − 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.228i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.973 - 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0358479 + 0.309548i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0358479 + 0.309548i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.756 + 1.55i)T \) |
| 5 | \( 1 + (2.04 - 0.911i)T \) |
| good | 2 | \( 1 + (0.807 + 0.565i)T + (0.684 + 1.87i)T^{2} \) |
| 7 | \( 1 + (0.275 + 0.590i)T + (-4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (-0.890 - 1.06i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (2.93 + 4.19i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (1.18 + 4.41i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-0.00652 + 0.00376i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.70 + 3.12i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (0.586 + 3.32i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-3.73 + 1.35i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-1.60 + 0.430i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (1.90 + 0.335i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (0.929 - 10.6i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (-6.61 + 3.08i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (1.18 + 1.18i)T + 53iT^{2} \) |
| 59 | \( 1 + (7.77 + 6.52i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (9.02 + 3.28i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (12.3 - 8.62i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (-5.41 - 3.12i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-12.1 - 3.25i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (0.782 - 0.137i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-5.32 + 7.59i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (-5.89 - 10.2i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.18 + 0.190i)T + (95.5 + 16.8i)T^{2} \) |
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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
−12.34018426747945596985949780568, −11.66242996118856029321708916729, −10.70276007118990473181285667355, −9.793421063522779435989352933795, −8.259455037970384746915526221492, −7.46934509367377983912174892314, −6.19951052982535607044297889828, −4.77532396222222634028384981136, −2.55385394068299444124080298113, −0.39668214506513471027505743975,
3.65877239709568437522668466808, 4.53332679927720834314835498430, 6.23753452994443877741462803028, 7.56157444695615710023108617287, 8.723622249396977840226790973349, 9.336553255974260730625421702083, 10.60599403864305692040279792152, 11.97499181534877526566432030718, 12.23829729064309691311350835362, 13.89369432515262843351612906004
