| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.984339223\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.984339223\) |
| \(L(\frac{3}{2})\) |
|
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
−8.680793167786295495888010418476, −7.85319280877638891237365304189, −7.42943471548663245901568718916, −5.90146246209421822764600605418, −5.19850069062180786906636299708, −4.88566444041661240710512766735, −4.07703968286016338613687044129, −3.34295539286390593000120855129, −2.28142057707166968120550968062, −0.73883349326847767033329887607,
0.929774698465912865937039803849, 1.74604155948059439970619927236, 2.76025410015650624852009261624, 3.22820115964361499363371137809, 5.02099572391018612786206727927, 5.39971682296916219527416646012, 6.32270382216909185750192352011, 6.98655850796622317295133755512, 7.74186960524477865956665733110, 8.004444230865431478009377884839
