| L(s) = 1 | + (0.509 + 1.23i)2-s + (−2.23 + 1.49i)3-s + (1.57 − 1.57i)4-s + (−1.59 − 0.317i)5-s + (−2.97 − 1.98i)6-s + (−8.69 + 1.72i)7-s + (7.66 + 3.17i)8-s + (−0.690 + 1.66i)9-s + (−0.422 − 2.12i)10-s + (4.60 − 6.88i)11-s + (−1.16 + 5.85i)12-s + (−16.1 − 16.1i)13-s + (−6.56 − 9.81i)14-s + (4.02 − 1.66i)15-s + 2.13i·16-s + ⋯ |
| L(s) = 1 | + (0.254 + 0.615i)2-s + (−0.743 + 0.496i)3-s + (0.393 − 0.393i)4-s + (−0.318 − 0.0634i)5-s + (−0.495 − 0.330i)6-s + (−1.24 + 0.247i)7-s + (0.957 + 0.396i)8-s + (−0.0767 + 0.185i)9-s + (−0.0422 − 0.212i)10-s + (0.418 − 0.626i)11-s + (−0.0970 + 0.488i)12-s + (−1.24 − 1.24i)13-s + (−0.468 − 0.701i)14-s + (0.268 − 0.111i)15-s + 0.133i·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.204016 - 0.258036i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.204016 - 0.258036i\) |
| \(L(2)\) |
|
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 | \( 1 + (-0.509 - 1.23i)T + (-2.82 + 2.82i)T^{2} \) |
| 3 | \( 1 + (2.23 - 1.49i)T + (3.44 - 8.31i)T^{2} \) |
| 5 | \( 1 + (1.59 + 0.317i)T + (23.0 + 9.56i)T^{2} \) |
| 7 | \( 1 + (8.69 - 1.72i)T + (45.2 - 18.7i)T^{2} \) |
| 11 | \( 1 + (-4.60 + 6.88i)T + (-46.3 - 111. i)T^{2} \) |
| 13 | \( 1 + (16.1 + 16.1i)T + 169iT^{2} \) |
| 19 | \( 1 + (0.500 + 1.20i)T + (-255. + 255. i)T^{2} \) |
| 23 | \( 1 + (22.2 + 14.8i)T + (202. + 488. i)T^{2} \) |
| 29 | \( 1 + (3.22 - 16.2i)T + (-776. - 321. i)T^{2} \) |
| 31 | \( 1 + (7.63 + 11.4i)T + (-367. + 887. i)T^{2} \) |
| 37 | \( 1 + (-16.0 + 10.7i)T + (523. - 1.26e3i)T^{2} \) |
| 41 | \( 1 + (-25.9 + 5.15i)T + (1.55e3 - 643. i)T^{2} \) |
| 43 | \( 1 + (9.06 - 21.8i)T + (-1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (26.8 + 26.8i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-10.8 - 26.1i)T + (-1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (22.0 + 9.12i)T + (2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-10.0 - 50.6i)T + (-3.43e3 + 1.42e3i)T^{2} \) |
| 67 | \( 1 - 70.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (87.2 - 58.2i)T + (1.92e3 - 4.65e3i)T^{2} \) |
| 73 | \( 1 + (34.9 + 6.94i)T + (4.92e3 + 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-4.03 + 6.03i)T + (-2.38e3 - 5.76e3i)T^{2} \) |
| 83 | \( 1 + (124. - 51.5i)T + (4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (38.6 - 38.6i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (15.4 - 77.7i)T + (-8.69e3 - 3.60e3i)T^{2} \) |
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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
−11.27501373638799286463263854219, −10.30214245313086543496212734996, −9.777419098629696503886758542291, −8.218823250543109614293089323758, −7.19101318702304538497823679849, −6.02330687237319773672924260994, −5.58162279127412754607979535650, −4.32192793708708641088058971470, −2.69681567059159154270571899868, −0.15065673483562743311085327421,
1.89381922226179256985075045699, 3.39359601594996889810975845954, 4.40634112170145046079974243834, 6.13606864879284655558581027417, 6.93860195291389678594108224195, 7.57427625339657218359166553660, 9.395470261976038419184136134360, 10.02494972144774136179643204894, 11.37784048164885476962034284434, 11.90090501710408130406070489696
