| L(s) = 1 | + 2i·2-s + 5.48i·3-s − 4·4-s − 10.9·6-s − 11.1i·7-s − 8i·8-s − 21.1·9-s − 21.9i·12-s + 22.2·14-s + 16·16-s − 42.2i·18-s + 61.0·21-s − 43.4i·23-s + 43.9·24-s − 66.6i·27-s + 44.4i·28-s + ⋯ |
| L(s) = 1 | + i·2-s + 1.82i·3-s − 4-s − 1.82·6-s − 1.58i·7-s − i·8-s − 2.34·9-s − 1.82i·12-s + 1.58·14-s + 16-s − 2.34i·18-s + 2.90·21-s − 1.89i·23-s + 1.82·24-s − 2.46i·27-s + 1.58i·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5893524537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5893524537\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2iT \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 5.48iT - 9T^{2} \) |
| 7 | \( 1 + 11.1iT - 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 + 43.4iT - 529T^{2} \) |
| 29 | \( 1 + 44.2T + 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 + 1.36e3T^{2} \) |
| 41 | \( 1 + 31.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 61.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 90.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 16.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 116iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 25.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 177.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 9.40e3T^{2} \) |
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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
−10.49504736791848907082431231366, −9.825248725030789013058932268937, −9.013513893099874192445745858267, −8.110248902730072314021312147522, −7.05488512804339152614364565433, −5.94247577423965593998414446763, −4.80232983611303413909266922288, −4.22858445346649028926870213167, −3.40872844728459214327175332530, −0.24002887859890979783760861728,
1.49906860654222495911093478563, 2.28121858908477957536326392648, 3.32550567490889787268131066507, 5.36483320337268677595760134125, 5.87782194402547518766285042381, 7.21615581741381449835262250241, 8.171725562383702631711224669051, 8.859887030165751889882033309133, 9.705472196546992065923075764546, 11.36616858573263128120706163855
