| L(s) = 1 | + (1.55 + 0.455i)2-s + (−1.46 + 1.68i)3-s + (0.519 + 0.334i)4-s + (0.460 − 3.20i)5-s + (−3.04 + 1.95i)6-s + (−0.513 − 1.12i)7-s + (−1.46 − 1.68i)8-s + (−0.284 − 1.97i)9-s + (2.17 − 4.76i)10-s + (0.732 − 0.215i)11-s + (−1.32 + 0.389i)12-s + (1.24 − 2.72i)13-s + (−0.284 − 1.97i)14-s + (4.73 + 5.46i)15-s + (−2.01 − 4.41i)16-s + (4.40 − 2.83i)17-s + ⋯ |
| L(s) = 1 | + (1.09 + 0.322i)2-s + (−0.845 + 0.975i)3-s + (0.259 + 0.167i)4-s + (0.205 − 1.43i)5-s + (−1.24 + 0.798i)6-s + (−0.194 − 0.424i)7-s + (−0.517 − 0.597i)8-s + (−0.0948 − 0.659i)9-s + (0.687 − 1.50i)10-s + (0.221 − 0.0648i)11-s + (−0.382 + 0.112i)12-s + (0.345 − 0.756i)13-s + (−0.0760 − 0.529i)14-s + (1.22 + 1.41i)15-s + (−0.504 − 1.10i)16-s + (1.06 − 0.686i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.708 + 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.708 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.48531 - 0.613363i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.48531 - 0.613363i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 \) |
| good | 2 | \( 1 + (-1.55 - 0.455i)T + (1.68 + 1.08i)T^{2} \) |
| 3 | \( 1 + (1.46 - 1.68i)T + (-0.426 - 2.96i)T^{2} \) |
| 5 | \( 1 + (-0.460 + 3.20i)T + (-4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (0.513 + 1.12i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (-0.732 + 0.215i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-1.24 + 2.72i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-4.40 + 2.83i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (1.68 + 1.08i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (2.52 - 1.62i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-4.39 - 5.06i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (0.460 + 3.20i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (0.778 - 5.41i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 2.23T + 47T^{2} \) |
| 53 | \( 1 + (3.51 + 7.70i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (1.02 - 2.24i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (7.16 + 8.27i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-6.94 - 2.03i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (7.44 + 2.18i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-13.0 - 8.36i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (-2.88 + 6.31i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-1.88 - 13.1i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-1.00 + 1.15i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (0.610 - 4.24i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−10.75411575415944662244488198455, −9.843306088852274132416855173501, −9.233455318915339406126364314966, −8.069887029653167109039521090074, −6.62275011110836415809605369812, −5.53072283146010222571962511834, −5.16258865044353444057420035222, −4.37954959355617894612649066301, −3.43858791272788125037823849868, −0.76195252838947469036014382491,
1.93064368649967139945652901033, 3.09583011535354888894607633982, 4.15813363166576713798659033991, 5.74444027686624805875868931506, 6.11413062933943001760226175972, 6.90919605484471532041398764172, 7.974366800887262670659036975163, 9.323279662169746991417635559692, 10.53974648044234234408900489018, 11.27087559894897441494697128771
