L(s) = 1 | + (−1.59 − 0.516i)2-s + (−1.82 − 2.51i)3-s + (0.643 + 0.467i)4-s + (1.60 + 4.94i)6-s + (−1.81 + 2.49i)7-s + (1.18 + 1.62i)8-s + (−2.06 + 6.34i)9-s + (−2.14 − 2.53i)11-s − 2.47i·12-s + (3.21 + 1.04i)13-s + (4.17 − 3.03i)14-s + (−1.53 − 4.71i)16-s + (0.0986 − 0.0320i)17-s + (6.55 − 9.02i)18-s + (−1.64 + 1.19i)19-s + ⋯ |
L(s) = 1 | + (−1.12 − 0.365i)2-s + (−1.05 − 1.45i)3-s + (0.321 + 0.233i)4-s + (0.655 + 2.01i)6-s + (−0.686 + 0.944i)7-s + (0.418 + 0.575i)8-s + (−0.687 + 2.11i)9-s + (−0.645 − 0.763i)11-s − 0.714i·12-s + (0.891 + 0.289i)13-s + (1.11 − 0.811i)14-s + (−0.383 − 1.17i)16-s + (0.0239 − 0.00777i)17-s + (1.54 − 2.12i)18-s + (−0.376 + 0.273i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.279i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 - 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.251970 + 0.0359793i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.251970 + 0.0359793i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 + (2.14 + 2.53i)T \) |
good | 2 | \( 1 + (1.59 + 0.516i)T + (1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (1.82 + 2.51i)T + (-0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 + (1.81 - 2.49i)T + (-2.16 - 6.65i)T^{2} \) |
| 13 | \( 1 + (-3.21 - 1.04i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.0986 + 0.0320i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (1.64 - 1.19i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + 2.23iT - 23T^{2} \) |
| 29 | \( 1 + (-5.25 - 3.81i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (2.80 - 8.63i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (0.195 - 0.269i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-3.18 + 2.31i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 7.39iT - 43T^{2} \) |
| 47 | \( 1 + (3.89 + 5.36i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (0.295 + 0.0961i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-9.07 - 6.59i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-1.04 - 3.22i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 - 12.8iT - 67T^{2} \) |
| 71 | \( 1 + (0.824 + 2.53i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (4.40 - 6.05i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-0.746 + 2.29i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (0.136 - 0.0442i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 + (-5.63 - 1.83i)T + (78.4 + 57.0i)T^{2} \) |
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
−11.79594732797488659990929394813, −10.99985132266784793077685410299, −10.27682192929672562313367954325, −8.776598765219473170111317172309, −8.319858912965702667245011840506, −7.04792484194483106315754580066, −6.10877847248546054936742929526, −5.26752996778268166758473860732, −2.63065969834142656175601998238, −1.21886181991094288477321325147,
0.36825281984796111780433870576, 3.71273802658606982898957988850, 4.54742154298861734941299038495, 5.94165465378993097743618170474, 6.93555776477470955758535567267, 8.122323646357608242517026533834, 9.393919716904845556764595870139, 9.921440404672162367554415921997, 10.57001676211508710428368447083, 11.27076592555928459900474227478