L(s) = 1 | + (−0.5 − 0.866i)2-s + (1.19 − 1.25i)3-s + (−0.499 + 0.866i)4-s + 0.465i·5-s + (−1.68 − 0.405i)6-s + (1.06 + 2.42i)7-s + 0.999·8-s + (−0.151 − 2.99i)9-s + (0.403 − 0.232i)10-s + (−1.39 − 2.41i)11-s + (0.490 + 1.66i)12-s + (0.799 − 3.51i)13-s + (1.56 − 2.13i)14-s + (0.584 + 0.555i)15-s + (−0.5 − 0.866i)16-s + (0.508 − 0.880i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.689 − 0.724i)3-s + (−0.249 + 0.433i)4-s + 0.208i·5-s + (−0.687 − 0.165i)6-s + (0.400 + 0.916i)7-s + 0.353·8-s + (−0.0504 − 0.998i)9-s + (0.127 − 0.0736i)10-s + (−0.421 − 0.729i)11-s + (0.141 + 0.479i)12-s + (0.221 − 0.975i)13-s + (0.419 − 0.569i)14-s + (0.150 + 0.143i)15-s + (−0.125 − 0.216i)16-s + (0.123 − 0.213i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0606 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0606 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.04260 - 1.10793i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.04260 - 1.10793i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (-1.19 + 1.25i)T \) |
| 7 | \( 1 + (-1.06 - 2.42i)T \) |
| 13 | \( 1 + (-0.799 + 3.51i)T \) |
good | 5 | \( 1 - 0.465iT - 5T^{2} \) |
| 11 | \( 1 + (1.39 + 2.41i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.508 + 0.880i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.817 + 1.41i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-7.14 + 4.12i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.01 + 2.31i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 1.94T + 31T^{2} \) |
| 37 | \( 1 + (-4.51 + 2.60i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (9.35 - 5.40i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.24 - 5.62i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 0.277iT - 47T^{2} \) |
| 53 | \( 1 - 8.99iT - 53T^{2} \) |
| 59 | \( 1 + (-8.80 - 5.08i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (11.2 + 6.49i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.308 - 0.178i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.01 - 8.68i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 1.46T + 73T^{2} \) |
| 79 | \( 1 + 4.98T + 79T^{2} \) |
| 83 | \( 1 - 12.2iT - 83T^{2} \) |
| 89 | \( 1 + (-10.1 + 5.85i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7.82 - 13.5i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.69369984381087273950890832245, −9.557971314103607446313485356733, −8.610893765166049478823963176212, −8.262098790611389425121390196788, −7.18833970851360606233487626655, −6.06236618706675490372464041182, −4.87232245267600213646955510799, −3.08984590783691390450970385605, −2.68486297479196999357339190714, −1.04170532032895753722465875113,
1.64632373162003313645831880878, 3.42171540111774582720426937124, 4.58110605207742662181940607307, 5.17715524018139112217753073412, 6.83995437334290847915786906845, 7.47501973512877116489481919218, 8.472060082713839555643387471416, 9.137142371359329156240250064456, 10.08200211256348427221297024449, 10.64943653827618072045681720974