| L(s) = 1 | − 4.27·5-s + 4.77i·7-s + 2.15i·11-s − 0.274·17-s + 4.35i·19-s + 8.71i·23-s + 13.2·25-s − 20.4i·35-s − 7.40i·43-s + 9.07i·47-s − 15.8·49-s − 9.19i·55-s − 3.72·61-s − 16.8·73-s − 10.2·77-s + ⋯ |
| L(s) = 1 | − 1.91·5-s + 1.80i·7-s + 0.648i·11-s − 0.0666·17-s + 0.999i·19-s + 1.81i·23-s + 2.65·25-s − 3.45i·35-s − 1.12i·43-s + 1.32i·47-s − 2.26·49-s − 1.23i·55-s − 0.476·61-s − 1.96·73-s − 1.17·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5147108378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5147108378\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
| good | 5 | \( 1 + 4.27T + 5T^{2} \) |
| 7 | \( 1 - 4.77iT - 7T^{2} \) |
| 11 | \( 1 - 2.15iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 0.274T + 17T^{2} \) |
| 23 | \( 1 - 8.71iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 7.40iT - 43T^{2} \) |
| 47 | \( 1 - 9.07iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 3.72T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 16.8T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 8.71iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.042108588132669311601872200220, −8.503243153787703328133084607152, −7.72652724054027929547864129782, −7.29022018769136603956781098860, −6.13535881065668641361519627083, −5.36295418401984817467550898969, −4.52415439635498938938854221339, −3.63064630154108936922242544463, −2.89791288848788442322312227279, −1.66607707720509697197046227194,
0.22315468211531304619483423828, 0.909996858983702918857335973493, 2.89330830437438205743543173680, 3.67587612221259471276879034555, 4.35682705620146562975115692051, 4.81735904721928808625906052292, 6.41675867384110287868133242206, 7.04910710671994780260937190658, 7.57170661324314634916581399459, 8.298295047762810795814621712560
