| L(s) = 1 | + 3i·5-s − 3.46i·11-s − 5.19i·13-s − 1.73·17-s + 4i·19-s + 6·23-s − 4·25-s + 3i·29-s − 10.3·31-s + 5.19i·37-s − 6.92·41-s + 2i·43-s + 6·47-s − 7·49-s + 6i·53-s + ⋯ |
| L(s) = 1 | + 1.34i·5-s − 1.04i·11-s − 1.44i·13-s − 0.420·17-s + 0.917i·19-s + 1.25·23-s − 0.800·25-s + 0.557i·29-s − 1.86·31-s + 0.854i·37-s − 1.08·41-s + 0.304i·43-s + 0.875·47-s − 49-s + 0.824i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.040442729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.040442729\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 3iT - 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 5.19iT - 13T^{2} \) |
| 17 | \( 1 + 1.73T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 3iT - 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 - 5.19iT - 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 6.92iT - 59T^{2} \) |
| 61 | \( 1 - 5.19iT - 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 13T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 10.3iT - 83T^{2} \) |
| 89 | \( 1 + 5.19T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.465329039850857472336869454569, −7.61646368902048937434264832357, −7.13060616782306697179590748148, −6.28926010492357299212555330101, −5.71582808478787352527320841656, −4.98047286829512682282735535615, −3.62101131754935313046802073182, −3.25466066739399267364538386651, −2.51596892523142798079160866314, −1.18142786946936707521433109560,
0.28371653185576898654626990594, 1.60189991808548534936640317830, 2.18942165913010360933706020078, 3.58818417341402796367213936923, 4.46056944612244463177172618268, 4.87191980603383368789362154465, 5.55090651791796245284618831930, 6.81802943609464534456166163683, 7.00453354780815373019477167339, 8.041983114660543579096820625147
