| L(s) = 1 | + (−1.31 + 0.979i)2-s + (1.61 − 0.613i)3-s + (0.198 − 0.662i)4-s + (0.0338 + 0.580i)5-s + (−1.53 + 2.39i)6-s + (−2.61 − 0.405i)7-s + (−0.734 − 2.01i)8-s + (2.24 − 1.98i)9-s + (−0.613 − 0.730i)10-s + (0.00477 − 0.00725i)11-s + (−0.0853 − 1.19i)12-s + (3.84 + 2.86i)13-s + (3.83 − 2.02i)14-s + (0.411 + 0.919i)15-s + (4.09 + 2.69i)16-s + (0.850 + 4.82i)17-s + ⋯ |
| L(s) = 1 | + (−0.930 + 0.692i)2-s + (0.935 − 0.354i)3-s + (0.0991 − 0.331i)4-s + (0.0151 + 0.259i)5-s + (−0.624 + 0.977i)6-s + (−0.988 − 0.153i)7-s + (−0.259 − 0.713i)8-s + (0.748 − 0.662i)9-s + (−0.193 − 0.231i)10-s + (0.00143 − 0.00218i)11-s + (−0.0246 − 0.344i)12-s + (1.06 + 0.794i)13-s + (1.02 − 0.542i)14-s + (0.106 + 0.237i)15-s + (1.02 + 0.673i)16-s + (0.206 + 1.16i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.280 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.280 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.905761 + 0.678692i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.905761 + 0.678692i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.61 + 0.613i)T \) |
| 7 | \( 1 + (2.61 + 0.405i)T \) |
| good | 2 | \( 1 + (1.31 - 0.979i)T + (0.573 - 1.91i)T^{2} \) |
| 5 | \( 1 + (-0.0338 - 0.580i)T + (-4.96 + 0.580i)T^{2} \) |
| 11 | \( 1 + (-0.00477 + 0.00725i)T + (-4.35 - 10.1i)T^{2} \) |
| 13 | \( 1 + (-3.84 - 2.86i)T + (3.72 + 12.4i)T^{2} \) |
| 17 | \( 1 + (-0.850 - 4.82i)T + (-15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (-3.18 - 0.561i)T + (17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (-1.50 - 6.36i)T + (-20.5 + 10.3i)T^{2} \) |
| 29 | \( 1 + (4.17 + 1.80i)T + (19.9 + 21.0i)T^{2} \) |
| 31 | \( 1 + (-2.21 - 9.36i)T + (-27.7 + 13.9i)T^{2} \) |
| 37 | \( 1 + (-7.60 + 6.37i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (10.5 + 1.22i)T + (39.8 + 9.45i)T^{2} \) |
| 43 | \( 1 + (0.810 + 0.533i)T + (17.0 + 39.4i)T^{2} \) |
| 47 | \( 1 + (-3.53 - 0.837i)T + (42.0 + 21.0i)T^{2} \) |
| 53 | \( 1 + 10.3iT - 53T^{2} \) |
| 59 | \( 1 + (-2.82 - 1.41i)T + (35.2 + 47.3i)T^{2} \) |
| 61 | \( 1 + (-2.41 + 0.721i)T + (50.9 - 33.5i)T^{2} \) |
| 67 | \( 1 + (4.93 + 0.576i)T + (65.1 + 15.4i)T^{2} \) |
| 71 | \( 1 + (4.35 - 11.9i)T + (-54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (13.4 + 2.36i)T + (68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-0.948 - 1.27i)T + (-22.6 + 75.6i)T^{2} \) |
| 83 | \( 1 + (-7.58 + 0.886i)T + (80.7 - 19.1i)T^{2} \) |
| 89 | \( 1 + (2.23 + 1.87i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (6.56 + 13.0i)T + (-57.9 + 77.8i)T^{2} \) |
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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
−10.47619800071772042304848173192, −9.717236657067165901429309009431, −8.967080416855112939855866367175, −8.406242488301126420275964532273, −7.34077431270640861216631854385, −6.80512505560478823810648055691, −5.94530643450170952653368861790, −3.83932830388405577174824518155, −3.25332797177365875971622681180, −1.37548996769210985842443986472,
0.926451400335284101611071625782, 2.62463090112006057164014178812, 3.27792173122221742514259627726, 4.79049502000323578908103558378, 5.99238570083577856944863708364, 7.35077936992237360185600870289, 8.346267638405832428336135769587, 8.993970018288567177748717950624, 9.648004737940803796264843525863, 10.28215178750063379784348505312
