| L(s) = 1 | + (−0.669 + 0.743i)2-s + (0.169 + 1.60i)3-s + (−0.104 − 0.994i)4-s + (−1.11 − 1.93i)5-s + (−1.30 − 0.951i)6-s + (−0.5 − 2.59i)7-s + (0.809 + 0.587i)8-s + (0.373 − 0.0794i)9-s + (2.18 + 0.464i)10-s + (2.79 + 0.593i)11-s + (1.58 − 0.336i)12-s + (0.263 − 0.812i)13-s + (2.26 + 1.36i)14-s + (2.92 − 2.12i)15-s + (−0.978 + 0.207i)16-s + (5.21 − 2.32i)17-s + ⋯ |
| L(s) = 1 | + (−0.473 + 0.525i)2-s + (0.0976 + 0.929i)3-s + (−0.0522 − 0.497i)4-s + (−0.499 − 0.866i)5-s + (−0.534 − 0.388i)6-s + (−0.188 − 0.981i)7-s + (0.286 + 0.207i)8-s + (0.124 − 0.0264i)9-s + (0.691 + 0.147i)10-s + (0.841 + 0.178i)11-s + (0.456 − 0.0971i)12-s + (0.0732 − 0.225i)13-s + (0.605 + 0.365i)14-s + (0.755 − 0.549i)15-s + (−0.244 + 0.0519i)16-s + (1.26 − 0.563i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.03250 + 0.0438369i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03250 + 0.0438369i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.669 - 0.743i)T \) |
| 5 | \( 1 + (1.11 + 1.93i)T \) |
| 7 | \( 1 + (0.5 + 2.59i)T \) |
| good | 3 | \( 1 + (-0.169 - 1.60i)T + (-2.93 + 0.623i)T^{2} \) |
| 11 | \( 1 + (-2.79 - 0.593i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-0.263 + 0.812i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-5.21 + 2.32i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-0.627 + 5.96i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (4.99 - 5.55i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (-4.42 + 3.21i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.69 + 0.754i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (2.56 - 0.544i)T + (33.8 - 15.0i)T^{2} \) |
| 41 | \( 1 + (-3.59 + 11.0i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 2.23T + 43T^{2} \) |
| 47 | \( 1 + (-4.35 - 1.93i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (0.442 + 4.21i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (-6.90 - 7.67i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (-2.51 + 2.79i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (0.913 - 0.406i)T + (44.8 - 49.7i)T^{2} \) |
| 71 | \( 1 + (-1.57 + 1.14i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (7.45 + 1.58i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (8.17 + 3.63i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-8.28 - 6.01i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (11.4 - 12.7i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (-5.47 + 3.97i)T + (29.9 - 92.2i)T^{2} \) |
| 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
−11.37000202606855455479294003113, −10.20447871428950403952498237790, −9.637035376137132495936479943091, −8.868935901518659063571030378579, −7.71870637998703602599005091704, −6.98339248047934901435999496157, −5.47331705499269407380791018758, −4.44964579301033497774120832530, −3.65367139258815347316627854860, −0.966614844692007552819694259470,
1.55488461948852048825723722498, 2.85158070675131371508215934399, 4.02012877990432601758428045700, 6.02531952229862431753809975289, 6.75014692465821840708195711418, 7.927335518483197722522852679050, 8.425643045441408518778175363831, 9.811288621272322477391489566624, 10.47567964109174806930833678564, 11.89152701079841700854976388675
