| L(s) = 1 | + (1.28 − 0.594i)2-s + (−0.989 − 0.142i)3-s + (1.29 − 1.52i)4-s + (−0.412 − 1.40i)5-s + (−1.35 + 0.405i)6-s + (−1.74 − 2.01i)7-s + (0.755 − 2.72i)8-s + (0.959 + 0.281i)9-s + (−1.36 − 1.55i)10-s + (0.913 + 0.587i)11-s + (−1.49 + 1.32i)12-s + (−0.741 + 0.856i)13-s + (−3.43 − 1.54i)14-s + (0.208 + 1.44i)15-s + (−0.650 − 3.94i)16-s + (0.484 − 0.221i)17-s + ⋯ |
| L(s) = 1 | + (0.907 − 0.420i)2-s + (−0.571 − 0.0821i)3-s + (0.647 − 0.762i)4-s + (−0.184 − 0.628i)5-s + (−0.553 + 0.165i)6-s + (−0.659 − 0.761i)7-s + (0.266 − 0.963i)8-s + (0.319 + 0.0939i)9-s + (−0.431 − 0.492i)10-s + (0.275 + 0.177i)11-s + (−0.432 + 0.382i)12-s + (−0.205 + 0.237i)13-s + (−0.918 − 0.413i)14-s + (0.0537 + 0.374i)15-s + (−0.162 − 0.986i)16-s + (0.117 − 0.0536i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.165 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05946 - 1.25262i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05946 - 1.25262i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.28 + 0.594i)T \) |
| 3 | \( 1 + (0.989 + 0.142i)T \) |
| 23 | \( 1 + (-4.14 - 2.41i)T \) |
| good | 5 | \( 1 + (0.412 + 1.40i)T + (-4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (1.74 + 2.01i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-0.913 - 0.587i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (0.741 - 0.856i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.484 + 0.221i)T + (11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-1.39 + 3.05i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-3.62 - 7.92i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (1.03 - 0.148i)T + (29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (2.79 - 9.50i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-1.71 + 0.502i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.818 + 5.69i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 6.12iT - 47T^{2} \) |
| 53 | \( 1 + (0.804 - 0.696i)T + (7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-6.08 - 5.27i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (5.19 - 0.746i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (2.68 - 1.72i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-5.64 - 8.79i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-1.57 + 3.43i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-10.5 + 12.2i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (15.1 + 4.43i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-9.94 - 1.42i)T + (85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (2.79 + 9.53i)T + (-81.6 + 52.4i)T^{2} \) |
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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
−11.81935822099678628931674331637, −10.85588230284942223744877850380, −10.04286985316285558158371639029, −8.985263215308144955843749628148, −7.22551080974785885187544260695, −6.61139969716779481436651525054, −5.24433770134237343073785156934, −4.45300870933811262808927232643, −3.17116373745018265014074389479, −1.11257317248882935605384007411,
2.66132501412801339946546625654, 3.80475323836042986284745267992, 5.20656691982311553836701296601, 6.12156704784662828643703217773, 6.87928340116719065587561469912, 7.992621284553492066914494181754, 9.309541286895055228887973145917, 10.56165072070013649669813910577, 11.41507267528819365313607036070, 12.28172224590419889299451843458
