| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0637i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.97023 - 0.0947052i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.97023 - 0.0947052i\) |
| \(L(2)\) |
|
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
−11.61579462062124618518299413519, −10.79288979217123016539272685436, −9.719198136029969802018755459252, −8.765338850846465669539859110393, −7.909590912162879551583583344917, −6.79537476072619861586772208431, −5.12385534255144355463142861080, −4.14366190833273379763728192610, −2.86611936223132083108706107916, −1.87701805353127618837541109247,
1.65227769947951261012965343758, 2.66181990325694410033452488817, 4.85021181486652900020967426910, 5.39197606041761939586060674921, 6.94345602345346785481453072298, 7.73545274288220290509379377889, 8.178141258418015786906723433324, 9.802062454260289109895958101052, 10.59553967810344077862694770602, 11.54826342042948259532498332483
