| L(s) = 1 | + (1.12 − 1.65i)2-s + (1.09 − 1.33i)3-s + (−1.44 − 3.72i)4-s + (2.44 + 8.07i)5-s + (−0.968 − 3.32i)6-s + (−2.13 + 10.7i)7-s + (−7.79 − 1.82i)8-s + (−0.585 − 2.94i)9-s + (16.0 + 5.07i)10-s + (2.17 + 0.214i)11-s + (−6.58 − 2.15i)12-s + (7.91 + 2.40i)13-s + (15.2 + 15.6i)14-s + (13.4 + 5.59i)15-s + (−11.8 + 10.8i)16-s + (3.38 + 8.18i)17-s + ⋯ |
| L(s) = 1 | + (0.564 − 0.825i)2-s + (0.366 − 0.446i)3-s + (−0.362 − 0.932i)4-s + (0.489 + 1.61i)5-s + (−0.161 − 0.554i)6-s + (−0.304 + 1.53i)7-s + (−0.973 − 0.227i)8-s + (−0.0650 − 0.326i)9-s + (1.60 + 0.507i)10-s + (0.197 + 0.0194i)11-s + (−0.548 − 0.179i)12-s + (0.609 + 0.184i)13-s + (1.09 + 1.11i)14-s + (0.899 + 0.372i)15-s + (−0.737 + 0.675i)16-s + (0.199 + 0.481i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0696i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0696i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.48620 + 0.0866601i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.48620 + 0.0866601i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.12 + 1.65i)T \) |
| 3 | \( 1 + (-1.09 + 1.33i)T \) |
| good | 5 | \( 1 + (-2.44 - 8.07i)T + (-20.7 + 13.8i)T^{2} \) |
| 7 | \( 1 + (2.13 - 10.7i)T + (-45.2 - 18.7i)T^{2} \) |
| 11 | \( 1 + (-2.17 - 0.214i)T + (118. + 23.6i)T^{2} \) |
| 13 | \( 1 + (-7.91 - 2.40i)T + (140. + 93.8i)T^{2} \) |
| 17 | \( 1 + (-3.38 - 8.18i)T + (-204. + 204. i)T^{2} \) |
| 19 | \( 1 + (-5.80 + 3.10i)T + (200. - 300. i)T^{2} \) |
| 23 | \( 1 + (-29.2 - 19.5i)T + (202. + 488. i)T^{2} \) |
| 29 | \( 1 + (20.9 - 2.05i)T + (824. - 164. i)T^{2} \) |
| 31 | \( 1 + (-24.2 + 24.2i)T - 961iT^{2} \) |
| 37 | \( 1 + (3.98 + 2.12i)T + (760. + 1.13e3i)T^{2} \) |
| 41 | \( 1 + (-15.9 - 10.6i)T + (643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (27.9 + 34.0i)T + (-360. + 1.81e3i)T^{2} \) |
| 47 | \( 1 + (30.5 + 73.7i)T + (-1.56e3 + 1.56e3i)T^{2} \) |
| 53 | \( 1 + (-61.5 - 6.05i)T + (2.75e3 + 548. i)T^{2} \) |
| 59 | \( 1 + (20.2 + 66.6i)T + (-2.89e3 + 1.93e3i)T^{2} \) |
| 61 | \( 1 + (64.9 - 79.1i)T + (-725. - 3.64e3i)T^{2} \) |
| 67 | \( 1 + (-88.3 - 72.5i)T + (875. + 4.40e3i)T^{2} \) |
| 71 | \( 1 + (-20.7 - 4.12i)T + (4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (82.1 - 16.3i)T + (4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-11.6 + 28.0i)T + (-4.41e3 - 4.41e3i)T^{2} \) |
| 83 | \( 1 + (36.4 + 68.2i)T + (-3.82e3 + 5.72e3i)T^{2} \) |
| 89 | \( 1 + (44.9 + 67.3i)T + (-3.03e3 + 7.31e3i)T^{2} \) |
| 97 | \( 1 + (-97.3 - 97.3i)T + 9.40e3iT^{2} \) |
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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
−11.39234915784418189872808867281, −10.31996418063073018420318361728, −9.469757008608972886978866898879, −8.666725097992895801696847914769, −7.06883012043266400440661846308, −6.18023260520830498825729547025, −5.50256881096108976203257604337, −3.54886945361727979613865998744, −2.79112524050184064519282020972, −1.86877424125920510847937658678,
0.939962938568223033148479267884, 3.32142689973859451344993670593, 4.41769797347148718024886434398, 5.00785807967801166686971391768, 6.23715267001603428925224065846, 7.38105743232628993379777012178, 8.332353827214585445093179841768, 9.110793088622864678435841145096, 9.901626424180000475866968266346, 11.12480918701125609909115763410
