| L(s) = 1 | − 2.12·3-s + (2.02 − 0.952i)5-s − i·7-s + 1.50·9-s − 5.19i·11-s − 6.02·13-s + (−4.29 + 2.02i)15-s + 2.84i·17-s + 1.52i·19-s + 2.12i·21-s + 4.80i·23-s + (3.18 − 3.85i)25-s + 3.18·27-s − 3.04i·29-s − 5.62·31-s + ⋯ |
| L(s) = 1 | − 1.22·3-s + (0.904 − 0.426i)5-s − 0.377i·7-s + 0.500·9-s − 1.56i·11-s − 1.67·13-s + (−1.10 + 0.521i)15-s + 0.690i·17-s + 0.350i·19-s + 0.462i·21-s + 1.00i·23-s + (0.636 − 0.770i)25-s + 0.612·27-s − 0.565i·29-s − 1.01·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.120i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2361502075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2361502075\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.02 + 0.952i)T \) |
| 7 | \( 1 + iT \) |
| good | 3 | \( 1 + 2.12T + 3T^{2} \) |
| 11 | \( 1 + 5.19iT - 11T^{2} \) |
| 13 | \( 1 + 6.02T + 13T^{2} \) |
| 17 | \( 1 - 2.84iT - 17T^{2} \) |
| 19 | \( 1 - 1.52iT - 19T^{2} \) |
| 23 | \( 1 - 4.80iT - 23T^{2} \) |
| 29 | \( 1 + 3.04iT - 29T^{2} \) |
| 31 | \( 1 + 5.62T + 31T^{2} \) |
| 37 | \( 1 + 4.69T + 37T^{2} \) |
| 41 | \( 1 - 5.14T + 41T^{2} \) |
| 43 | \( 1 + 5.28T + 43T^{2} \) |
| 47 | \( 1 - 6.17iT - 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + 0.438iT - 59T^{2} \) |
| 61 | \( 1 + 0.0169iT - 61T^{2} \) |
| 67 | \( 1 + 3.18T + 67T^{2} \) |
| 71 | \( 1 + 5.46T + 71T^{2} \) |
| 73 | \( 1 - 6.46iT - 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 6.30T + 89T^{2} \) |
| 97 | \( 1 + 3.58iT - 97T^{2} \) |
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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
−9.605630242414947328334575851051, −8.668576419655378912438823245231, −7.67271896585540425543894566939, −6.63990513242064425281951347360, −5.77005179515613495775832534602, −5.43260317108536958673690998704, −4.40696137283330433269437190755, −3.01393119849091899175907591346, −1.51043142698452376391827403889, −0.11681079495325514783120892976,
1.89761339223388663017382954200, 2.80277571510824551552089755583, 4.74224833582811275957779313160, 5.04272569674119299818642554662, 5.99531967625293970568286480089, 6.98969001126051096319324599516, 7.26364280355705571954504796387, 8.868877923515095394744292613575, 9.736596991525132520504217764413, 10.17267723493528080473981689747
