| L(s) = 1 | + (1.15 − 2.79i)2-s + (0.451 − 0.675i)3-s + (−3.65 − 3.65i)4-s + (1.56 + 7.87i)5-s + (−1.36 − 2.04i)6-s + (−1.53 + 7.70i)7-s + (−3.25 + 1.34i)8-s + (3.19 + 7.70i)9-s + (23.8 + 4.74i)10-s + (6.71 − 4.48i)11-s + (−4.11 + 0.818i)12-s + (−0.798 + 0.798i)13-s + (19.7 + 13.2i)14-s + (6.02 + 2.49i)15-s − 9.98i·16-s + ⋯ |
| L(s) = 1 | + (0.579 − 1.39i)2-s + (0.150 − 0.225i)3-s + (−0.913 − 0.913i)4-s + (0.313 + 1.57i)5-s + (−0.227 − 0.340i)6-s + (−0.218 + 1.10i)7-s + (−0.407 + 0.168i)8-s + (0.354 + 0.856i)9-s + (2.38 + 0.474i)10-s + (0.610 − 0.407i)11-s + (−0.342 + 0.0682i)12-s + (−0.0614 + 0.0614i)13-s + (1.41 + 0.943i)14-s + (0.401 + 0.166i)15-s − 0.624i·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.499i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.31997 - 0.621142i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.31997 - 0.621142i\) |
| \(L(2)\) |
|
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 | \( 1 + (-1.15 + 2.79i)T + (-2.82 - 2.82i)T^{2} \) |
| 3 | \( 1 + (-0.451 + 0.675i)T + (-3.44 - 8.31i)T^{2} \) |
| 5 | \( 1 + (-1.56 - 7.87i)T + (-23.0 + 9.56i)T^{2} \) |
| 7 | \( 1 + (1.53 - 7.70i)T + (-45.2 - 18.7i)T^{2} \) |
| 11 | \( 1 + (-6.71 + 4.48i)T + (46.3 - 111. i)T^{2} \) |
| 13 | \( 1 + (0.798 - 0.798i)T - 169iT^{2} \) |
| 19 | \( 1 + (1.07 - 2.59i)T + (-255. - 255. i)T^{2} \) |
| 23 | \( 1 + (-4.76 - 7.13i)T + (-202. + 488. i)T^{2} \) |
| 29 | \( 1 + (-3.01 + 0.599i)T + (776. - 321. i)T^{2} \) |
| 31 | \( 1 + (10.6 + 7.13i)T + (367. + 887. i)T^{2} \) |
| 37 | \( 1 + (13.1 - 19.6i)T + (-523. - 1.26e3i)T^{2} \) |
| 41 | \( 1 + (-4.26 + 21.4i)T + (-1.55e3 - 643. i)T^{2} \) |
| 43 | \( 1 + (-8.89 - 21.4i)T + (-1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (-55.6 + 55.6i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-22.9 + 55.5i)T + (-1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-25.3 + 10.5i)T + (2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (35.7 + 7.11i)T + (3.43e3 + 1.42e3i)T^{2} \) |
| 67 | \( 1 + 117. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (58.8 - 88.1i)T + (-1.92e3 - 4.65e3i)T^{2} \) |
| 73 | \( 1 + (11.9 + 59.8i)T + (-4.92e3 + 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-79.2 + 52.9i)T + (2.38e3 - 5.76e3i)T^{2} \) |
| 83 | \( 1 + (-109. - 45.4i)T + (4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (61.4 + 61.4i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (24.1 - 4.79i)T + (8.69e3 - 3.60e3i)T^{2} \) |
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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
−11.50567415286976536029093451904, −10.73506030125031977688471330560, −10.06169150089782547801005194903, −9.012379211658827220093369989468, −7.52167668833242516092837151179, −6.43590751111012632568255654116, −5.27269778821752892829824568305, −3.70463449658537879715572717078, −2.71666767119809817943131720705, −1.93800407195500003428923560234,
1.09202952574998482065273963983, 3.99618775196988598351790157129, 4.51324079129767245144318481059, 5.65339323585441527003929292980, 6.71829922327067009528276152869, 7.55004971965215327558042242733, 8.749444245662944581352157399674, 9.366978871836361339943454050340, 10.51344042787556109788724711189, 12.17457886282845662265515998489
