L(s) = 1 | + (−0.309 + 0.951i)2-s + (−0.417 − 0.303i)3-s + (−0.809 − 0.587i)4-s + (−1.20 + 1.88i)5-s + (0.417 − 0.303i)6-s + 0.938·7-s + (0.809 − 0.587i)8-s + (−0.844 − 2.59i)9-s + (−1.41 − 1.73i)10-s + (0.530 − 1.63i)11-s + (0.159 + 0.491i)12-s + (1.63 + 5.04i)13-s + (−0.290 + 0.893i)14-s + (1.07 − 0.418i)15-s + (0.309 + 0.951i)16-s + (−3.14 + 2.28i)17-s + ⋯ |
L(s) = 1 | + (−0.218 + 0.672i)2-s + (−0.241 − 0.175i)3-s + (−0.404 − 0.293i)4-s + (−0.540 + 0.841i)5-s + (0.170 − 0.123i)6-s + 0.354·7-s + (0.286 − 0.207i)8-s + (−0.281 − 0.866i)9-s + (−0.447 − 0.547i)10-s + (0.159 − 0.492i)11-s + (0.0460 + 0.141i)12-s + (0.454 + 1.39i)13-s + (−0.0775 + 0.238i)14-s + (0.277 − 0.108i)15-s + (0.0772 + 0.237i)16-s + (−0.762 + 0.553i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0588195 - 0.318766i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0588195 - 0.318766i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 - 0.951i)T \) |
| 5 | \( 1 + (1.20 - 1.88i)T \) |
| 19 | \( 1 + (0.809 - 0.587i)T \) |
good | 3 | \( 1 + (0.417 + 0.303i)T + (0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 - 0.938T + 7T^{2} \) |
| 11 | \( 1 + (-0.530 + 1.63i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-1.63 - 5.04i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (3.14 - 2.28i)T + (5.25 - 16.1i)T^{2} \) |
| 23 | \( 1 + (1.52 - 4.69i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (3.43 + 2.49i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (6.13 - 4.45i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (2.97 + 9.16i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-0.997 - 3.06i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 6.94T + 43T^{2} \) |
| 47 | \( 1 + (6.53 + 4.75i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (4.78 + 3.47i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.990 - 3.04i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.48 + 4.55i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (13.0 - 9.47i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (3.47 + 2.52i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.389 + 1.19i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (5.75 + 4.17i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-0.751 + 0.545i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (4.22 - 13.0i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-14.5 - 10.5i)T + (29.9 + 92.2i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.64091482627603033625673285726, −9.370493926115232376991634480607, −8.856002082768401413737592583759, −7.87610996273946010197923037265, −6.99761679704720214366539595400, −6.40756404810519103180293810929, −5.64620647961221705039815237773, −4.18773802755267451280577464253, −3.53484315778426311241199931805, −1.76835458475845072515046489793,
0.16672427920181844417528887555, 1.71951961509836229155051641952, 3.02372528351399062588488324120, 4.32709259304703515818630304370, 4.90685822923304896054331430834, 5.84084007288513558729311240525, 7.37127046175535473837009286243, 8.099968957753898103475499993735, 8.716883552415785079360616241640, 9.620087198384670690166301978915