| 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
−11.27087559894897441494697128771, −10.53974648044234234408900489018, −9.323279662169746991417635559692, −7.974366800887262670659036975163, −6.90919605484471532041398764172, −6.11413062933943001760226175972, −5.74444027686624805875868931506, −4.15813363166576713798659033991, −3.09583011535354888894607633982, −1.93064368649967139945652901033,
0.76195252838947469036014382491, 3.43858791272788125037823849868, 4.37954959355617894612649066301, 5.16258865044353444057420035222, 5.53072283146010222571962511834, 6.62275011110836415809605369812, 8.069887029653167109039521090074, 9.233455318915339406126364314966, 9.843306088852274132416855173501, 10.75411575415944662244488198455
