| L(s) = 1 | − 2.23·5-s + 9.06i·7-s − 4.28i·11-s − 9.41·13-s − 18·17-s − 36.2i·19-s + 22.9i·23-s + 5.00·25-s + 44.8·29-s − 35.2i·31-s − 20.2i·35-s + 6.58·37-s − 52.2·41-s − 28.8i·43-s − 90.1i·47-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.29i·7-s − 0.389i·11-s − 0.724·13-s − 1.05·17-s − 1.90i·19-s + 0.996i·23-s + 0.200·25-s + 1.54·29-s − 1.13i·31-s − 0.579i·35-s + 0.177·37-s − 1.27·41-s − 0.670i·43-s − 1.91i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5475001624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5475001624\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
| good | 7 | \( 1 - 9.06iT - 49T^{2} \) |
| 11 | \( 1 + 4.28iT - 121T^{2} \) |
| 13 | \( 1 + 9.41T + 169T^{2} \) |
| 17 | \( 1 + 18T + 289T^{2} \) |
| 19 | \( 1 + 36.2iT - 361T^{2} \) |
| 23 | \( 1 - 22.9iT - 529T^{2} \) |
| 29 | \( 1 - 44.8T + 841T^{2} \) |
| 31 | \( 1 + 35.2iT - 961T^{2} \) |
| 37 | \( 1 - 6.58T + 1.36e3T^{2} \) |
| 41 | \( 1 + 52.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + 28.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 90.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 52.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 17.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 50.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 33.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 20.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 91.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 42.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 22.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 47.6T + 7.92e3T^{2} \) |
| 97 | \( 1 + 160.T + 9.40e3T^{2} \) |
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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
−9.777001380695223774437175726667, −8.946271507237837294037211348633, −8.416027863237256524158062977361, −7.24826059709875262024955942401, −6.42833519498707328197249739787, −5.30689146390029485069246659303, −4.55015624217656607730695593699, −3.08818026969010172978020151958, −2.19923826177031711785418606456, −0.19381442377873155802409579160,
1.38179626340448308843889577599, 2.97123484421082209794274889103, 4.20874931189570728910642451186, 4.73259953585066063772744938246, 6.28100680247349600879873704327, 7.04360105212120725429100136339, 7.85879684291984121109542862582, 8.627020564082968737503518958073, 9.931575698883027979388856226885, 10.37103574611051485180370108688
