| L(s) = 1 | + (0.841 − 2.03i)2-s + (−0.134 − 0.0897i)3-s + (−0.590 − 0.590i)4-s + (−5.26 + 1.04i)5-s + (−0.295 + 0.197i)6-s + (6.12 + 1.21i)7-s + (6.42 − 2.66i)8-s + (−3.43 − 8.29i)9-s + (−2.30 + 11.5i)10-s + (−8.11 − 12.1i)11-s + (0.0263 + 0.132i)12-s + (4.79 − 4.79i)13-s + (7.62 − 11.4i)14-s + (0.801 + 0.331i)15-s − 18.6i·16-s + ⋯ |
| L(s) = 1 | + (0.420 − 1.01i)2-s + (−0.0447 − 0.0299i)3-s + (−0.147 − 0.147i)4-s + (−1.05 + 0.209i)5-s + (−0.0492 + 0.0329i)6-s + (0.874 + 0.174i)7-s + (0.803 − 0.332i)8-s + (−0.381 − 0.921i)9-s + (−0.230 + 1.15i)10-s + (−0.737 − 1.10i)11-s + (0.00219 + 0.0110i)12-s + (0.369 − 0.369i)13-s + (0.544 − 0.815i)14-s + (0.0534 + 0.0221i)15-s − 1.16i·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.694 + 0.719i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.694 + 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.670869 - 1.58098i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.670869 - 1.58098i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + (-0.841 + 2.03i)T + (-2.82 - 2.82i)T^{2} \) |
| 3 | \( 1 + (0.134 + 0.0897i)T + (3.44 + 8.31i)T^{2} \) |
| 5 | \( 1 + (5.26 - 1.04i)T + (23.0 - 9.56i)T^{2} \) |
| 7 | \( 1 + (-6.12 - 1.21i)T + (45.2 + 18.7i)T^{2} \) |
| 11 | \( 1 + (8.11 + 12.1i)T + (-46.3 + 111. i)T^{2} \) |
| 13 | \( 1 + (-4.79 + 4.79i)T - 169iT^{2} \) |
| 19 | \( 1 + (-9.56 + 23.0i)T + (-255. - 255. i)T^{2} \) |
| 23 | \( 1 + (10.8 - 7.27i)T + (202. - 488. i)T^{2} \) |
| 29 | \( 1 + (6.44 + 32.3i)T + (-776. + 321. i)T^{2} \) |
| 31 | \( 1 + (0.674 - 1.00i)T + (-367. - 887. i)T^{2} \) |
| 37 | \( 1 + (-38.6 - 25.8i)T + (523. + 1.26e3i)T^{2} \) |
| 41 | \( 1 + (-30.8 - 6.13i)T + (1.55e3 + 643. i)T^{2} \) |
| 43 | \( 1 + (-27.8 - 67.3i)T + (-1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (10.4 - 10.4i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-1.97 + 4.77i)T + (-1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (26.1 - 10.8i)T + (2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-16.1 + 81.0i)T + (-3.43e3 - 1.42e3i)T^{2} \) |
| 67 | \( 1 - 44.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (48.1 + 32.1i)T + (1.92e3 + 4.65e3i)T^{2} \) |
| 73 | \( 1 + (1.32 - 0.262i)T + (4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (16.5 + 24.7i)T + (-2.38e3 + 5.76e3i)T^{2} \) |
| 83 | \( 1 + (-62.2 - 25.7i)T + (4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-90.1 - 90.1i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (-13.1 - 66.3i)T + (-8.69e3 + 3.60e3i)T^{2} \) |
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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
−11.40884694220982706124680724367, −10.89348466083484461359907605561, −9.509720940111464369136750675208, −8.177288414519561164863095740399, −7.63393677008903512244260183252, −6.09993435715801482111361110636, −4.72291548590796907469686016681, −3.59858309325577400361813647426, −2.74221739737522678143024151509, −0.77012092068317367549961549757,
1.92942334238449468301746031517, 4.11941355957671650738742634086, 4.90082305545277564676240471278, 5.82272112874805401520492217689, 7.46996548896238260426759513302, 7.63157144538600198525681008312, 8.571999195642934351661747524198, 10.28771015337888283509443044197, 11.01929642318122678250439441119, 11.92669197314994438149454242961
