| L(s) = 1 | − 1.67i·3-s − 0.714i·5-s + 1.81i·7-s + 0.180·9-s − 1.90i·11-s − 0.684·13-s − 1.20·15-s + (−2.75 − 3.06i)17-s − 8.58·19-s + 3.04·21-s + 2.35i·23-s + 4.48·25-s − 5.34i·27-s − 1.67i·29-s + 9.13i·31-s + ⋯ |
| L(s) = 1 | − 0.969i·3-s − 0.319i·5-s + 0.684i·7-s + 0.0602·9-s − 0.573i·11-s − 0.189·13-s − 0.309·15-s + (−0.668 − 0.743i)17-s − 1.97·19-s + 0.663·21-s + 0.490i·23-s + 0.897·25-s − 1.02i·27-s − 0.310i·29-s + 1.64i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.668 - 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1737141960\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1737141960\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (2.75 + 3.06i)T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 1.67iT - 3T^{2} \) |
| 5 | \( 1 + 0.714iT - 5T^{2} \) |
| 7 | \( 1 - 1.81iT - 7T^{2} \) |
| 11 | \( 1 + 1.90iT - 11T^{2} \) |
| 13 | \( 1 + 0.684T + 13T^{2} \) |
| 19 | \( 1 + 8.58T + 19T^{2} \) |
| 23 | \( 1 - 2.35iT - 23T^{2} \) |
| 29 | \( 1 + 1.67iT - 29T^{2} \) |
| 31 | \( 1 - 9.13iT - 31T^{2} \) |
| 37 | \( 1 - 2.45iT - 37T^{2} \) |
| 41 | \( 1 + 5.79iT - 41T^{2} \) |
| 43 | \( 1 + 0.853T + 43T^{2} \) |
| 47 | \( 1 + 5.83T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 61 | \( 1 + 0.855iT - 61T^{2} \) |
| 67 | \( 1 + 2.72T + 67T^{2} \) |
| 71 | \( 1 + 1.42iT - 71T^{2} \) |
| 73 | \( 1 - 1.06iT - 73T^{2} \) |
| 79 | \( 1 - 10.5iT - 79T^{2} \) |
| 83 | \( 1 + 8.47T + 83T^{2} \) |
| 89 | \( 1 + 17.7T + 89T^{2} \) |
| 97 | \( 1 + 14.7iT - 97T^{2} \) |
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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
−8.129659916517187372083535824001, −7.05570224728641572904107562650, −6.66719964800816877878850844470, −5.90618208487186515191217863961, −5.00139754532534485823173887986, −4.28487952362475227706635639699, −3.04603631697617245945203723610, −2.22246632285741374467065403752, −1.35552054857911612039488397413, −0.04755343441751497349305316115,
1.67387074427870070910277046798, 2.68057753567736383340580505713, 3.82207582981896894377832841587, 4.36904894655475621126741759136, 4.80362348497580466604134511648, 6.08348999973035929804108897076, 6.66871291928480040420606332531, 7.39634557859069178437587942635, 8.288799546533697624402862166683, 8.948023024857265189616781002592
