| L(s) = 1 | + (−1.39 − 0.221i)2-s + (0.252 + 0.495i)3-s + (1.90 + 0.618i)4-s + (3.68 − 3.38i)5-s + (−0.243 − 0.748i)6-s + (7.20 + 7.20i)7-s + (−2.52 − 1.28i)8-s + (5.10 − 7.03i)9-s + (−5.88 + 3.91i)10-s + (4.56 − 3.31i)11-s + (0.174 + 1.09i)12-s + (−22.6 + 3.58i)13-s + (−8.46 − 11.6i)14-s + (2.60 + 0.969i)15-s + (3.23 + 2.35i)16-s + (−8.10 + 15.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.698 − 0.110i)2-s + (0.0842 + 0.165i)3-s + (0.475 + 0.154i)4-s + (0.736 − 0.676i)5-s + (−0.0405 − 0.124i)6-s + (1.02 + 1.02i)7-s + (−0.315 − 0.160i)8-s + (0.567 − 0.781i)9-s + (−0.588 + 0.391i)10-s + (0.415 − 0.301i)11-s + (0.0145 + 0.0915i)12-s + (−1.74 + 0.275i)13-s + (−0.604 − 0.832i)14-s + (0.173 + 0.0646i)15-s + (0.202 + 0.146i)16-s + (−0.476 + 0.936i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.114i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.993 + 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.999401 - 0.0575259i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.999401 - 0.0575259i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 + 0.221i)T \) |
| 5 | \( 1 + (-3.68 + 3.38i)T \) |
| good | 3 | \( 1 + (-0.252 - 0.495i)T + (-5.29 + 7.28i)T^{2} \) |
| 7 | \( 1 + (-7.20 - 7.20i)T + 49iT^{2} \) |
| 11 | \( 1 + (-4.56 + 3.31i)T + (37.3 - 115. i)T^{2} \) |
| 13 | \( 1 + (22.6 - 3.58i)T + (160. - 52.2i)T^{2} \) |
| 17 | \( 1 + (8.10 - 15.9i)T + (-169. - 233. i)T^{2} \) |
| 19 | \( 1 + (13.6 - 4.41i)T + (292. - 212. i)T^{2} \) |
| 23 | \( 1 + (-4.05 + 25.6i)T + (-503. - 163. i)T^{2} \) |
| 29 | \( 1 + (14.8 + 4.81i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (-10.7 - 33.0i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (-2.42 - 15.3i)T + (-1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (37.3 + 27.1i)T + (519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (31.8 - 31.8i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-8.16 + 4.16i)T + (1.29e3 - 1.78e3i)T^{2} \) |
| 53 | \( 1 + (12.8 + 25.2i)T + (-1.65e3 + 2.27e3i)T^{2} \) |
| 59 | \( 1 + (23.3 - 32.0i)T + (-1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-11.5 + 8.37i)T + (1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 + (-24.8 + 48.7i)T + (-2.63e3 - 3.63e3i)T^{2} \) |
| 71 | \( 1 + (23.8 - 73.5i)T + (-4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-15.4 + 97.7i)T + (-5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (-110. - 35.8i)T + (5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (13.6 + 6.95i)T + (4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + (-43.0 - 59.2i)T + (-2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-67.9 + 34.6i)T + (5.53e3 - 7.61e3i)T^{2} \) |
| 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
−15.20935027304311441256707977550, −14.51749523455517956978949238308, −12.63394173340149683164700075419, −11.93691094403066206093042204959, −10.31510509759679074499698276851, −9.182790474143254956498347405418, −8.377933847127410057428261932703, −6.48479763387985505483261145100, −4.80874172120921449497613091236, −1.97603508183990244614847532532,
2.05943713332830541585227747716, 4.89887582646527228188531339257, 7.03468414370944004064625223804, 7.65735531674739523606844995067, 9.564622810730615022895000296112, 10.45449487280145365538554843042, 11.50255254597743280499151270177, 13.29073639700178520011458863247, 14.29958560905537748655100835822, 15.22504456111064616123658567265
