| L(s) = 1 | + (−0.312 − 1.37i)2-s + (−0.547 + 3.10i)3-s + (−1.80 + 0.862i)4-s + (1.46 + 1.23i)5-s + (4.45 − 0.215i)6-s + (1.58 − 0.917i)7-s + (1.75 + 2.21i)8-s + (−6.53 − 2.38i)9-s + (1.23 − 2.40i)10-s + (−2.10 − 1.21i)11-s + (−1.69 − 6.08i)12-s + (2.49 − 0.440i)13-s + (−1.76 − 1.90i)14-s + (−4.62 + 3.88i)15-s + (2.51 − 3.11i)16-s + (0.818 − 0.297i)17-s + ⋯ |
| L(s) = 1 | + (−0.220 − 0.975i)2-s + (−0.316 + 1.79i)3-s + (−0.902 + 0.431i)4-s + (0.655 + 0.550i)5-s + (1.81 − 0.0879i)6-s + (0.600 − 0.346i)7-s + (0.619 + 0.784i)8-s + (−2.17 − 0.793i)9-s + (0.391 − 0.761i)10-s + (−0.635 − 0.366i)11-s + (−0.487 − 1.75i)12-s + (0.692 − 0.122i)13-s + (−0.470 − 0.509i)14-s + (−1.19 + 1.00i)15-s + (0.628 − 0.777i)16-s + (0.198 − 0.0722i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.761990 + 0.225101i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.761990 + 0.225101i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.312 + 1.37i)T \) |
| 19 | \( 1 + (-4.35 + 0.0369i)T \) |
| good | 3 | \( 1 + (0.547 - 3.10i)T + (-2.81 - 1.02i)T^{2} \) |
| 5 | \( 1 + (-1.46 - 1.23i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-1.58 + 0.917i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.10 + 1.21i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.49 + 0.440i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.818 + 0.297i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (4.69 + 5.59i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (0.967 - 2.65i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-2.07 - 3.59i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 1.48iT - 37T^{2} \) |
| 41 | \( 1 + (1.43 + 0.252i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-5.26 + 6.27i)T + (-7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (0.855 - 2.34i)T + (-36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (1.18 + 1.41i)T + (-9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (6.43 - 2.34i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (8.58 - 7.20i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (2.59 + 0.945i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-6.12 - 5.13i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-2.02 + 11.4i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-0.908 + 5.15i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (8.19 - 4.73i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (6.18 - 1.09i)T + (83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (2.11 + 5.81i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−14.39193447130835282230666393976, −13.80875545902263965918899237196, −11.98706812950668687146504067933, −10.73614761190690216916212699285, −10.52928766767504844706868415710, −9.461871347814430463461731793400, −8.279019890160977658441550036293, −5.67597484690463575688542454921, −4.45898219318946145642921758202, −3.07627921115643240427239682067,
1.56569846403822636688391946995, 5.33834178028767348443249002810, 6.08803531223674575594378867979, 7.51405262242820659611118673279, 8.185857720796239802180415839868, 9.538976156226998333027484819568, 11.40600212180341000246998303885, 12.62127278297792399949229490895, 13.49532848097851777539704139720, 14.05282683082582877258708902759
