| L(s) = 1 | + (0.00133 + 0.0152i)2-s + (−1.71 + 0.259i)3-s + (1.96 − 0.347i)4-s + (0.516 + 2.17i)5-s + (−0.00623 − 0.0256i)6-s + (−0.148 − 0.211i)7-s + (0.0158 + 0.0589i)8-s + (2.86 − 0.890i)9-s + (−0.0324 + 0.0107i)10-s + (1.49 + 4.11i)11-s + (−3.28 + 1.10i)12-s + (2.14 + 0.187i)13-s + (0.00301 − 0.00253i)14-s + (−1.44 − 3.59i)15-s + (3.75 − 1.36i)16-s + (0.456 − 1.70i)17-s + ⋯ |
| L(s) = 1 | + (0.000940 + 0.0107i)2-s + (−0.988 + 0.150i)3-s + (0.984 − 0.173i)4-s + (0.230 + 0.972i)5-s + (−0.00254 − 0.0104i)6-s + (−0.0559 − 0.0799i)7-s + (0.00558 + 0.0208i)8-s + (0.954 − 0.296i)9-s + (−0.0102 + 0.00339i)10-s + (0.451 + 1.23i)11-s + (−0.947 + 0.319i)12-s + (0.594 + 0.0519i)13-s + (0.000806 − 0.000676i)14-s + (−0.374 − 0.927i)15-s + (0.939 − 0.341i)16-s + (0.110 − 0.412i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.999802 + 0.307417i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.999802 + 0.307417i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.71 - 0.259i)T \) |
| 5 | \( 1 + (-0.516 - 2.17i)T \) |
| good | 2 | \( 1 + (-0.00133 - 0.0152i)T + (-1.96 + 0.347i)T^{2} \) |
| 7 | \( 1 + (0.148 + 0.211i)T + (-2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (-1.49 - 4.11i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-2.14 - 0.187i)T + (12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-0.456 + 1.70i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (4.91 + 2.83i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.74 + 3.32i)T + (7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (3.59 + 3.01i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.912 + 5.17i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-3.12 - 0.837i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-0.241 - 0.288i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-8.25 - 3.84i)T + (27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (3.28 - 2.29i)T + (16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (-8.15 + 8.15i)T - 53iT^{2} \) |
| 59 | \( 1 + (10.6 + 3.86i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (2.10 - 11.9i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-0.160 + 1.82i)T + (-65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-4.44 + 2.56i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (13.8 - 3.71i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (1.98 - 2.37i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-4.94 + 0.432i)T + (81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (-1.44 + 2.50i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.99 - 4.27i)T + (-62.3 - 74.3i)T^{2} \) |
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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
−13.14345231059330656070319323805, −12.00772656069716231928731929517, −11.25161026325097659759182014834, −10.45562683460791004767870065089, −9.641950736731205142476289833510, −7.51848572538072015606065024203, −6.64318417391124470863528482327, −5.94160013909297780420177589587, −4.18136179243150601731694339069, −2.16746532124694019446929528508,
1.52627555749593633901222829523, 3.91500703877155790579078562506, 5.73932246663363520889091744623, 6.20097373316244300640062250652, 7.72745718120405652634351466585, 8.867982170993105267899753330481, 10.40589998136125103439572774339, 11.17462459192928489934579014331, 12.14660508713317740360851621977, 12.77628300486435246785383185480
