L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 + 2.59i)7-s + 0.999·8-s + (−0.499 − 0.866i)10-s + (−3 − 5.19i)11-s − 4·13-s + (−2 − 1.73i)14-s + (−0.5 + 0.866i)16-s + (−1 + 1.73i)19-s + 0.999·20-s + 6·22-s + (−1.5 + 2.59i)23-s + (−0.499 − 0.866i)25-s + (2 − 3.46i)26-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.223 + 0.387i)5-s + (−0.188 + 0.981i)7-s + 0.353·8-s + (−0.158 − 0.273i)10-s + (−0.904 − 1.56i)11-s − 1.10·13-s + (−0.534 − 0.462i)14-s + (−0.125 + 0.216i)16-s + (−0.229 + 0.397i)19-s + 0.223·20-s + 1.27·22-s + (−0.312 + 0.541i)23-s + (−0.0999 − 0.173i)25-s + (0.392 − 0.679i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.5 - 2.59i)T \) |
good | 11 | \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 + 7T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + (-8 - 13.8i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1 - 1.73i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 3T + 83T^{2} \) |
| 89 | \( 1 + (7.5 - 12.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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
−10.13674555933449688595706986875, −9.362263137949630077746650162252, −8.330218493211894504475909875971, −7.85601007663382126392462740744, −6.67905207650146596867597715996, −5.78149055665161598359757250672, −5.10483093209606245564055100831, −3.48424463123868087787204580029, −2.34142976950219957224109029413, 0,
1.80840339071381180761949501197, 3.09052373664154797180614200692, 4.50626227222765011065380375675, 4.93265213906589917326214678358, 6.80833063567805711105378780383, 7.44645354959296084882293345592, 8.277608869902214080331965399769, 9.420465461694282824218267692750, 10.17197762304243319352463022947