L(s) = 1 | + (0.254 − 2.42i)2-s + (−3.84 − 0.818i)4-s + (3.77 − 0.396i)5-s + (0.273 + 0.246i)7-s + (−1.45 + 4.48i)8-s − 9.24i·10-s + (2.75 + 1.84i)11-s + (−0.385 − 0.866i)13-s + (0.665 − 0.599i)14-s + (3.30 + 1.47i)16-s + (2.77 − 2.01i)17-s + (4.05 + 1.31i)19-s + (−14.8 − 1.56i)20-s + (5.17 − 6.20i)22-s + (−4.30 − 2.48i)23-s + ⋯ |
L(s) = 1 | + (0.180 − 1.71i)2-s + (−1.92 − 0.409i)4-s + (1.68 − 0.177i)5-s + (0.103 + 0.0930i)7-s + (−0.515 + 1.58i)8-s − 2.92i·10-s + (0.830 + 0.557i)11-s + (−0.106 − 0.240i)13-s + (0.177 − 0.160i)14-s + (0.826 + 0.367i)16-s + (0.673 − 0.489i)17-s + (0.929 + 0.302i)19-s + (−3.32 − 0.349i)20-s + (1.10 − 1.32i)22-s + (−0.897 − 0.517i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.719673 - 2.08124i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.719673 - 2.08124i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + (-2.75 - 1.84i)T \) |
good | 2 | \( 1 + (-0.254 + 2.42i)T + (-1.95 - 0.415i)T^{2} \) |
| 5 | \( 1 + (-3.77 + 0.396i)T + (4.89 - 1.03i)T^{2} \) |
| 7 | \( 1 + (-0.273 - 0.246i)T + (0.731 + 6.96i)T^{2} \) |
| 13 | \( 1 + (0.385 + 0.866i)T + (-8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (-2.77 + 2.01i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-4.05 - 1.31i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (4.30 + 2.48i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.66 - 1.84i)T + (-3.03 - 28.8i)T^{2} \) |
| 31 | \( 1 + (-3.21 + 1.42i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (2.21 + 6.83i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (1.81 + 2.02i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (1.63 - 0.943i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.00428 + 0.0201i)T + (-42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (3.25 - 4.48i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (1.37 - 6.47i)T + (-53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (3.96 - 8.90i)T + (-40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (2.23 - 3.86i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.06 - 8.35i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (4.18 - 1.35i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (10.8 + 1.14i)T + (77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (8.38 + 3.73i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + 3.04iT - 89T^{2} \) |
| 97 | \( 1 + (-1.57 + 15.0i)T + (-94.8 - 20.1i)T^{2} \) |
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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
−10.01227953598626329566964725538, −9.386361644315387579441449147091, −8.701647994120389880788010138332, −7.20416524407369550738619553479, −5.93093552702309900529861285235, −5.22136233146373007737201547022, −4.18865226617555717026263430161, −2.98398572652778916230384923831, −2.03262503735435045687481056834, −1.20361100840009606270786374729,
1.59013871781476403636401569323, 3.33260137649721837462129451024, 4.70609346881987407918893332590, 5.57220592765045289023375848970, 6.18082564096992469711714671188, 6.73951642316776494920351282455, 7.79108194836004197090993873082, 8.628987512002492127030947261497, 9.547235623865241200876971847166, 9.893497928688817843288466816771