| L(s) = 1 | + 2.99i·3-s − 1.25i·5-s + (−2.36 + 1.18i)7-s − 5.94·9-s + (−3.16 − 1.00i)11-s + 13-s + 3.74·15-s − 5.49·17-s − 3.45·19-s + (−3.55 − 7.06i)21-s + 0.0769·23-s + 3.42·25-s − 8.79i·27-s + 1.19i·29-s − 10.2i·31-s + ⋯ |
| L(s) = 1 | + 1.72i·3-s − 0.560i·5-s + (−0.893 + 0.449i)7-s − 1.98·9-s + (−0.952 − 0.303i)11-s + 0.277·13-s + 0.967·15-s − 1.33·17-s − 0.791·19-s + (−0.775 − 1.54i)21-s + 0.0160·23-s + 0.685·25-s − 1.69i·27-s + 0.222i·29-s − 1.84i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8105213789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8105213789\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.36 - 1.18i)T \) |
| 11 | \( 1 + (3.16 + 1.00i)T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 2.99iT - 3T^{2} \) |
| 5 | \( 1 + 1.25iT - 5T^{2} \) |
| 17 | \( 1 + 5.49T + 17T^{2} \) |
| 19 | \( 1 + 3.45T + 19T^{2} \) |
| 23 | \( 1 - 0.0769T + 23T^{2} \) |
| 29 | \( 1 - 1.19iT - 29T^{2} \) |
| 31 | \( 1 + 10.2iT - 31T^{2} \) |
| 37 | \( 1 - 2.73T + 37T^{2} \) |
| 41 | \( 1 - 4.91T + 41T^{2} \) |
| 43 | \( 1 - 9.53iT - 43T^{2} \) |
| 47 | \( 1 + 0.917iT - 47T^{2} \) |
| 53 | \( 1 - 7.45T + 53T^{2} \) |
| 59 | \( 1 - 10.7iT - 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 + 9.82T + 67T^{2} \) |
| 71 | \( 1 - 16.6T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 - 7.54iT - 79T^{2} \) |
| 83 | \( 1 + 7.11T + 83T^{2} \) |
| 89 | \( 1 + 3.72iT - 89T^{2} \) |
| 97 | \( 1 + 7.07iT - 97T^{2} \) |
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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
−8.708799619398660746642996242678, −8.119968925571293888311688798035, −6.85806665667935586818608010064, −5.88960265755379003773829803311, −5.50987661638034212015167365039, −4.40333265999976728383218228563, −4.21072672549835169320135582712, −3.02250594908510478702885772928, −2.43400157292299818016614071173, −0.31961900290491830472924090820,
0.74770873893276980260487286175, 2.08104409347045353815209790698, 2.63100773197857938954854678077, 3.54614432389199607754331035678, 4.74420670666531891562437964376, 5.82534301153593515229514194751, 6.49694911490809057942040911538, 7.01389305801799064013519196004, 7.35433687617118366533030335103, 8.407534526865269030374884411527
