| L(s) = 1 | + (−0.0777 − 0.00680i)2-s + (−1.54 − 0.788i)3-s + (−1.96 − 0.346i)4-s + (0.114 + 0.0718i)6-s + (1.14 + 0.804i)7-s + (0.301 + 0.0807i)8-s + (1.75 + 2.43i)9-s + (−1.25 + 3.43i)11-s + (2.75 + 2.08i)12-s + (−0.326 − 3.73i)13-s + (−0.0838 − 0.0703i)14-s + (3.72 + 1.35i)16-s + (4.17 − 1.11i)17-s + (−0.120 − 0.201i)18-s + (−1.40 + 0.808i)19-s + ⋯ |
| L(s) = 1 | + (−0.0549 − 0.00481i)2-s + (−0.890 − 0.455i)3-s + (−0.981 − 0.173i)4-s + (0.0467 + 0.0293i)6-s + (0.434 + 0.304i)7-s + (0.106 + 0.0285i)8-s + (0.585 + 0.810i)9-s + (−0.377 + 1.03i)11-s + (0.795 + 0.600i)12-s + (−0.0906 − 1.03i)13-s + (−0.0224 − 0.0188i)14-s + (0.931 + 0.338i)16-s + (1.01 − 0.271i)17-s + (−0.0283 − 0.0473i)18-s + (−0.321 + 0.185i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.239 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.239 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.579707 - 0.454184i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.579707 - 0.454184i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.54 + 0.788i)T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (0.0777 + 0.00680i)T + (1.96 + 0.347i)T^{2} \) |
| 7 | \( 1 + (-1.14 - 0.804i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (1.25 - 3.43i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.326 + 3.73i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-4.17 + 1.11i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (1.40 - 0.808i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.27 + 4.67i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (-3.26 + 2.73i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.336 + 1.91i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (2.07 + 7.76i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (2.08 - 2.48i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.347 - 0.746i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-0.661 + 0.945i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (-7.50 + 7.50i)T - 53iT^{2} \) |
| 59 | \( 1 + (-9.85 + 3.58i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (1.72 + 9.76i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-14.7 + 1.28i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-8.12 - 4.69i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.0713 - 0.266i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (7.24 + 8.63i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (1.48 - 17.0i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (-6.77 - 11.7i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (12.5 - 5.86i)T + (62.3 - 74.3i)T^{2} \) |
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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
−10.12741143026469135043292850053, −9.834415578569730590264734384831, −8.295846073789354472611533442998, −7.893967903519704512625927035438, −6.72065007079807397105850901226, −5.49954035791701605145569680303, −5.10817658082820341761016621518, −4.01245521322469905794817801710, −2.17397414583837433014630913811, −0.57665717810707641534108718723,
1.12037495610579000663777122904, 3.42755179243139080965063520323, 4.29691131415629793427992537270, 5.19182162308766752335894843316, 5.95735551080234279128041818696, 7.16230689369667475375150585111, 8.233141465649350428823582934399, 8.997555539111953786399193226558, 9.965485626958989713048302001498, 10.54209242583094556517807107822
