| L(s) = 1 | + (1.23 − 3.80i)2-s + (6.17 + 18.9i)3-s + (−12.9 − 9.40i)4-s − 50.2·5-s + 79.8·6-s + (−60.1 − 43.7i)7-s + (−51.7 + 37.6i)8-s + (−126. + 91.6i)9-s + (−62.1 + 191. i)10-s + (−415. − 301. i)11-s + (98.7 − 303. i)12-s + (−197. − 607. i)13-s + (−240. + 174. i)14-s + (−310. − 955. i)15-s + (79.1 + 243. i)16-s + (−334. + 242. i)17-s + ⋯ |
| L(s) = 1 | + (0.218 − 0.672i)2-s + (0.395 + 1.21i)3-s + (−0.404 − 0.293i)4-s − 0.899·5-s + 0.906·6-s + (−0.464 − 0.337i)7-s + (−0.286 + 0.207i)8-s + (−0.519 + 0.377i)9-s + (−0.196 + 0.604i)10-s + (−1.03 − 0.751i)11-s + (0.197 − 0.609i)12-s + (−0.324 − 0.997i)13-s + (−0.328 + 0.238i)14-s + (−0.356 − 1.09i)15-s + (0.0772 + 0.237i)16-s + (−0.280 + 0.203i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.120i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0114837 - 0.189894i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0114837 - 0.189894i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.23 + 3.80i)T \) |
| 31 | \( 1 + (2.77e3 - 4.57e3i)T \) |
| good | 3 | \( 1 + (-6.17 - 18.9i)T + (-196. + 142. i)T^{2} \) |
| 5 | \( 1 + 50.2T + 3.12e3T^{2} \) |
| 7 | \( 1 + (60.1 + 43.7i)T + (5.19e3 + 1.59e4i)T^{2} \) |
| 11 | \( 1 + (415. + 301. i)T + (4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (197. + 607. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (334. - 242. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (20.2 - 62.4i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (2.02e3 - 1.46e3i)T + (1.98e6 - 6.12e6i)T^{2} \) |
| 29 | \( 1 + (1.28e3 - 3.94e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 37 | \( 1 - 4.91e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-6.00e3 + 1.84e4i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 + (-1.86e3 + 5.75e3i)T + (-1.18e8 - 8.64e7i)T^{2} \) |
| 47 | \( 1 + (-3.12e3 - 9.62e3i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (1.05e4 - 7.63e3i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-5.56e3 - 1.71e4i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + 9.97e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.88e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + (1.85e4 - 1.34e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (5.63e4 + 4.09e4i)T + (6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-2.42e4 + 1.76e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-2.11e4 + 6.51e4i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + (-8.48e4 - 6.16e4i)T + (1.72e9 + 5.31e9i)T^{2} \) |
| 97 | \( 1 + (4.23e3 + 3.07e3i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−13.42189266686405460752558078952, −12.30242385687837461465026098877, −10.84550849525937908321223971694, −10.29693527399914686012426798715, −9.003059848580940322951534093609, −7.73459516030039283389648462545, −5.40627261407274107777300328259, −3.97626464756684034536302832904, −3.09421724350862409206478402822, −0.07515829177805759200204653426,
2.39347964318324893579282226111, 4.40532685221393992955636605969, 6.31119108088616008075090274094, 7.45696214943180349950028580841, 8.083709997046615717665508603177, 9.610759152392415025750200793054, 11.61836155929371576839135604582, 12.60851432610380969094448299798, 13.32997208759815445668680503357, 14.53172889691532432650908710988
