L(s) = 1 | + (−1.57 + 0.729i)3-s + (0.115 + 0.200i)5-s + (0.230 − 0.399i)7-s + (1.93 − 2.29i)9-s + (−0.749 + 1.29i)11-s + (−1.07 − 1.85i)13-s + (−0.328 − 0.230i)15-s + 1.03·17-s − 2.94·19-s + (−0.0707 + 0.796i)21-s + (0.364 + 0.631i)23-s + (2.47 − 4.28i)25-s + (−1.36 + 5.01i)27-s + (2.33 − 4.04i)29-s + (−2.73 − 4.73i)31-s + ⋯ |
L(s) = 1 | + (−0.906 + 0.421i)3-s + (0.0518 + 0.0897i)5-s + (0.0872 − 0.151i)7-s + (0.644 − 0.764i)9-s + (−0.225 + 0.391i)11-s + (−0.296 − 0.514i)13-s + (−0.0847 − 0.0595i)15-s + 0.251·17-s − 0.675·19-s + (−0.0154 + 0.173i)21-s + (0.0760 + 0.131i)23-s + (0.494 − 0.856i)25-s + (−0.262 + 0.964i)27-s + (0.433 − 0.751i)29-s + (−0.491 − 0.851i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 + 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.640 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9549588766\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9549588766\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.57 - 0.729i)T \) |
good | 5 | \( 1 + (-0.115 - 0.200i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.230 + 0.399i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.749 - 1.29i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.07 + 1.85i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 1.03T + 17T^{2} \) |
| 19 | \( 1 + 2.94T + 19T^{2} \) |
| 23 | \( 1 + (-0.364 - 0.631i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.33 + 4.04i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.73 + 4.73i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2.30T + 37T^{2} \) |
| 41 | \( 1 + (1.84 + 3.18i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.41 - 4.19i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.40 + 9.36i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 + (-2.71 - 4.70i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.86 + 11.8i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.58 + 9.67i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 - 4.24T + 73T^{2} \) |
| 79 | \( 1 + (-6.23 + 10.8i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.62 + 6.28i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + (2.21 - 3.83i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.980993963915493423369430255494, −8.997002173784103643660290812298, −7.974457202464624824286517271152, −7.07354794148684091530794925168, −6.25371398199862357423261886698, −5.39032997789170454688536595780, −4.58531625488504077228810232068, −3.67177608412428627134701351854, −2.24598869766005400968423651884, −0.53653494771406744354910501270,
1.16514141071341200940931421130, 2.43351743115085670674798801664, 3.87745568459247231663700188686, 5.03385344364500313199096089336, 5.56896162321280411383281245494, 6.68682798739786860366563299025, 7.17016972425761290660073206260, 8.283992281442573966699567573556, 9.018026788306769025434460895982, 10.12957937868975153419604892823