| L(s) = 1 | + (−2.89 + 6.01i)2-s + (−13.0 − 0.977i)3-s + (−17.8 − 22.3i)4-s + (−7.18 + 23.2i)5-s + (43.6 − 75.6i)6-s + (24.2 − 14.0i)7-s + (81.7 − 18.6i)8-s + (89.2 + 13.4i)9-s + (−119. − 110. i)10-s + (−73.5 + 92.1i)11-s + (210. + 308. i)12-s + (199. − 185. i)13-s + (13.9 + 186. i)14-s + (116. − 296. i)15-s + (−22.8 + 99.9i)16-s + (−289. + 89.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.724 + 1.50i)2-s + (−1.44 − 0.108i)3-s + (−1.11 − 1.39i)4-s + (−0.287 + 0.931i)5-s + (1.21 − 2.10i)6-s + (0.495 − 0.286i)7-s + (1.27 − 0.291i)8-s + (1.10 + 0.166i)9-s + (−1.19 − 1.10i)10-s + (−0.607 + 0.761i)11-s + (1.46 + 2.14i)12-s + (1.18 − 1.09i)13-s + (0.0714 + 0.952i)14-s + (0.517 − 1.31i)15-s + (−0.0890 + 0.390i)16-s + (−1.00 + 0.309i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 + 0.503i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.863 + 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.157715 - 0.0426430i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.157715 - 0.0426430i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-478. + 1.78e3i)T \) |
| good | 2 | \( 1 + (2.89 - 6.01i)T + (-9.97 - 12.5i)T^{2} \) |
| 3 | \( 1 + (13.0 + 0.977i)T + (80.0 + 12.0i)T^{2} \) |
| 5 | \( 1 + (7.18 - 23.2i)T + (-516. - 352. i)T^{2} \) |
| 7 | \( 1 + (-24.2 + 14.0i)T + (1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (73.5 - 92.1i)T + (-3.25e3 - 1.42e4i)T^{2} \) |
| 13 | \( 1 + (-199. + 185. i)T + (2.13e3 - 2.84e4i)T^{2} \) |
| 17 | \( 1 + (289. - 89.3i)T + (6.90e4 - 4.70e4i)T^{2} \) |
| 19 | \( 1 + (46.1 + 306. i)T + (-1.24e5 + 3.84e4i)T^{2} \) |
| 23 | \( 1 + (373. + 952. i)T + (-2.05e5 + 1.90e5i)T^{2} \) |
| 29 | \( 1 + (-191. + 14.3i)T + (6.99e5 - 1.05e5i)T^{2} \) |
| 31 | \( 1 + (1.40e3 - 959. i)T + (3.37e5 - 8.59e5i)T^{2} \) |
| 37 | \( 1 + (15.4 + 8.93i)T + (9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 + (-715. - 344. i)T + (1.76e6 + 2.20e6i)T^{2} \) |
| 47 | \( 1 + (1.43e3 + 1.79e3i)T + (-1.08e6 + 4.75e6i)T^{2} \) |
| 53 | \( 1 + (2.23e3 + 2.07e3i)T + (5.89e5 + 7.86e6i)T^{2} \) |
| 59 | \( 1 + (-814. + 3.56e3i)T + (-1.09e7 - 5.25e6i)T^{2} \) |
| 61 | \( 1 + (-1.07e3 + 1.57e3i)T + (-5.05e6 - 1.28e7i)T^{2} \) |
| 67 | \( 1 + (2.03e3 - 306. i)T + (1.92e7 - 5.93e6i)T^{2} \) |
| 71 | \( 1 + (-1.85e3 - 729. i)T + (1.86e7 + 1.72e7i)T^{2} \) |
| 73 | \( 1 + (-441. - 475. i)T + (-2.12e6 + 2.83e7i)T^{2} \) |
| 79 | \( 1 + (800. + 1.38e3i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-951. + 1.26e4i)T + (-4.69e7 - 7.07e6i)T^{2} \) |
| 89 | \( 1 + (1.06e4 + 796. i)T + (6.20e7 + 9.35e6i)T^{2} \) |
| 97 | \( 1 + (4.03e3 - 5.05e3i)T + (-1.96e7 - 8.63e7i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.50895039481034388183633744786, −14.50745555088704259313507624523, −12.80954576447137457124167561816, −11.01054880259762858463915314197, −10.46701279512357747478672242892, −8.423479519863329765996030735949, −7.10389067712762070515628033790, −6.29879712500237837683692239267, −4.96756708495893990404846587317, −0.17339375418537740917602106869,
1.40453122087826638700597930242, 4.17689992881441536981845870305, 5.78258144367208086327299771982, 8.290690270025316159397524689447, 9.422213868472859886364457460414, 11.08239365049456508541196642116, 11.31946738298775760791343136410, 12.35106906303516498091964298096, 13.45403981539817639682632971587, 15.95557271007314627040543595613
