| L(s) = 1 | + (−0.826 − 0.563i)2-s + (−0.539 − 0.500i)3-s + (0.365 + 0.930i)4-s + (−2.65 + 0.817i)5-s + (0.163 + 0.717i)6-s + (0.222 − 0.974i)8-s + (−0.183 − 2.45i)9-s + (2.65 + 0.817i)10-s + (0.207 − 2.76i)11-s + (0.268 − 0.684i)12-s + (−3.14 + 1.51i)13-s + (1.83 + 0.885i)15-s + (−0.733 + 0.680i)16-s + (4.34 − 0.654i)17-s + (−1.22 + 2.12i)18-s + (2.26 + 3.92i)19-s + ⋯ |
| L(s) = 1 | + (−0.584 − 0.398i)2-s + (−0.311 − 0.288i)3-s + (0.182 + 0.465i)4-s + (−1.18 + 0.365i)5-s + (0.0668 + 0.292i)6-s + (0.0786 − 0.344i)8-s + (−0.0612 − 0.817i)9-s + (0.838 + 0.258i)10-s + (0.0624 − 0.833i)11-s + (0.0775 − 0.197i)12-s + (−0.872 + 0.419i)13-s + (0.474 + 0.228i)15-s + (−0.183 + 0.170i)16-s + (1.05 − 0.158i)17-s + (−0.289 + 0.501i)18-s + (0.520 + 0.901i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.432 - 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.432 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.371978 + 0.234254i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.371978 + 0.234254i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.826 + 0.563i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (0.539 + 0.500i)T + (0.224 + 2.99i)T^{2} \) |
| 5 | \( 1 + (2.65 - 0.817i)T + (4.13 - 2.81i)T^{2} \) |
| 11 | \( 1 + (-0.207 + 2.76i)T + (-10.8 - 1.63i)T^{2} \) |
| 13 | \( 1 + (3.14 - 1.51i)T + (8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + (-4.34 + 0.654i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (-2.26 - 3.92i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.18 + 0.480i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (-5.83 - 7.31i)T + (-6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (4.56 - 7.91i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.06 - 7.80i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (-0.176 + 0.773i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (-0.242 - 1.06i)T + (-38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (2.41 + 1.64i)T + (17.1 + 43.7i)T^{2} \) |
| 53 | \( 1 + (-3.04 - 7.76i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (-3.22 - 0.995i)T + (48.7 + 33.2i)T^{2} \) |
| 61 | \( 1 + (-0.144 + 0.367i)T + (-44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (-5.94 + 10.2i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.55 + 5.71i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (7.57 - 5.16i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (-2.66 - 4.62i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-9.31 - 4.48i)T + (51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (0.124 + 1.66i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−10.67080218691302787293578094728, −9.871797161337425683202015343588, −8.857031325243747699697441916652, −8.029372355432578832049542847667, −7.25367418513078841888956790156, −6.48043161543260096588962229584, −5.19165406661897701420505142478, −3.69918296676991242365703323478, −3.14194679212656133037044121413, −1.19688785260336256466038591286,
0.33566747162563456980491250275, 2.33095638134433380581785004018, 4.00640141159824570038411736967, 4.89049300885424125020778020029, 5.72751017130831322478895494215, 7.20425553381055309163598103900, 7.71448678311006868679469037472, 8.342543325160687009175205723416, 9.641815910631805234103679390632, 10.09368704280560449528278000922
