| L(s) = 1 | + i·3-s − 1.12i·5-s − 7-s − 9-s + 4.76i·11-s + 0.456i·13-s + 1.12·15-s + 0.415·17-s + 7.63i·19-s − i·21-s − 1.58·23-s + 3.72·25-s − i·27-s + 6.72i·29-s + 5.89·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.504i·5-s − 0.377·7-s − 0.333·9-s + 1.43i·11-s + 0.126i·13-s + 0.291·15-s + 0.100·17-s + 1.75i·19-s − 0.218i·21-s − 0.330·23-s + 0.745·25-s − 0.192i·27-s + 1.24i·29-s + 1.05·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0806 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0806 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.817191 + 0.885987i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.817191 + 0.885987i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 1.12iT - 5T^{2} \) |
| 11 | \( 1 - 4.76iT - 11T^{2} \) |
| 13 | \( 1 - 0.456iT - 13T^{2} \) |
| 17 | \( 1 - 0.415T + 17T^{2} \) |
| 19 | \( 1 - 7.63iT - 19T^{2} \) |
| 23 | \( 1 + 1.58T + 23T^{2} \) |
| 29 | \( 1 - 6.72iT - 29T^{2} \) |
| 31 | \( 1 - 5.89T + 31T^{2} \) |
| 37 | \( 1 - 5.89iT - 37T^{2} \) |
| 41 | \( 1 + 0.415T + 41T^{2} \) |
| 43 | \( 1 + 9.43iT - 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 - 7.63iT - 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 + 1.80iT - 61T^{2} \) |
| 67 | \( 1 - 8.09iT - 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + 3.34T + 73T^{2} \) |
| 79 | \( 1 - 4.83T + 79T^{2} \) |
| 83 | \( 1 + 5.53iT - 83T^{2} \) |
| 89 | \( 1 - 4.92T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−10.28932016128989123630810383501, −10.08821969118824940901067057829, −9.084384043901513228314544406795, −8.263586164551981682758355305099, −7.23637952681003013110710467551, −6.22021168595438117023328059903, −5.10447322469561774764906262652, −4.34742679918832510305077508914, −3.23297899597053526629589846514, −1.67812458116808409065748220128,
0.65598735063407581382633387420, 2.55423358264363743807354881153, 3.35151951317659564134683822012, 4.82513064607062105979023788244, 6.09070650266674010681489188302, 6.58212053187187790824094701773, 7.64377110455079001001612965512, 8.489318592901268744997481028118, 9.332771554494283214426506256042, 10.39437579334748864153724868126
