| L(s) = 1 | + (0.913 − 0.406i)2-s + (0.978 − 0.207i)3-s + (0.669 − 0.743i)4-s + (−0.305 − 2.90i)5-s + (0.809 − 0.587i)6-s + (−2.53 + 0.754i)7-s + (0.309 − 0.951i)8-s + (0.913 − 0.406i)9-s + (−1.46 − 2.53i)10-s + (3.12 − 1.12i)11-s + (0.499 − 0.866i)12-s + (0.928 + 0.674i)13-s + (−2.00 + 1.72i)14-s + (−0.904 − 2.78i)15-s + (−0.104 − 0.994i)16-s + (1.39 + 0.620i)17-s + ⋯ |
| L(s) = 1 | + (0.645 − 0.287i)2-s + (0.564 − 0.120i)3-s + (0.334 − 0.371i)4-s + (−0.136 − 1.30i)5-s + (0.330 − 0.239i)6-s + (−0.958 + 0.285i)7-s + (0.109 − 0.336i)8-s + (0.304 − 0.135i)9-s + (−0.462 − 0.801i)10-s + (0.940 − 0.339i)11-s + (0.144 − 0.249i)12-s + (0.257 + 0.187i)13-s + (−0.537 + 0.459i)14-s + (−0.233 − 0.718i)15-s + (−0.0261 − 0.248i)16-s + (0.337 + 0.150i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0937 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0937 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.64558 - 1.49796i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64558 - 1.49796i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.913 + 0.406i)T \) |
| 3 | \( 1 + (-0.978 + 0.207i)T \) |
| 7 | \( 1 + (2.53 - 0.754i)T \) |
| 11 | \( 1 + (-3.12 + 1.12i)T \) |
| good | 5 | \( 1 + (0.305 + 2.90i)T + (-4.89 + 1.03i)T^{2} \) |
| 13 | \( 1 + (-0.928 - 0.674i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.39 - 0.620i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (4.41 + 4.90i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-0.745 + 1.29i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.65 - 5.08i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.327 - 3.11i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-3.69 - 0.786i)T + (33.8 + 15.0i)T^{2} \) |
| 41 | \( 1 + (3.65 - 11.2i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 4.19T + 43T^{2} \) |
| 47 | \( 1 + (0.548 + 0.609i)T + (-4.91 + 46.7i)T^{2} \) |
| 53 | \( 1 + (1.29 - 12.3i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (-9.01 + 10.0i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-1.10 - 10.5i)T + (-59.6 + 12.6i)T^{2} \) |
| 67 | \( 1 + (-0.555 - 0.961i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.46 + 3.24i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-6.28 + 6.97i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (9.09 - 4.05i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (10.8 - 7.87i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-1.58 + 2.74i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (6.96 + 5.05i)T + (29.9 + 92.2i)T^{2} \) |
| 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
−11.00895211204655918119480686690, −9.771209464758655122131405352404, −8.976037725226927181230021164204, −8.459729342189448325844236639785, −6.92540181072782843454362649049, −6.12463629484336599080785114376, −4.85671595677614990342711311026, −3.95947577563340185983911709907, −2.81866150050788771673701128973, −1.18337956665492193533586178314,
2.34983085858869111871103028703, 3.53432506445444979385293023648, 4.03995269002626705810975386587, 5.86374124872713508709811120288, 6.66395871911974635731236395284, 7.31797667483135407707149917432, 8.393292175669119188131508039838, 9.655919920621552876181574297792, 10.31173634441655990353510201953, 11.26839934418698647123307475596
