| L(s) = 1 | + (−1.49 − 1.66i)5-s − 0.712i·7-s − 2.90·11-s − 6.88i·13-s + 0.433i·17-s + 7.42·19-s − 5.73i·23-s + (−0.545 + 4.97i)25-s + 10.5·29-s + 0.00620·31-s + (−1.18 + 1.06i)35-s + i·37-s + 3.79·41-s − 4.77i·43-s − 3.37i·47-s + ⋯ |
| L(s) = 1 | + (−0.667 − 0.744i)5-s − 0.269i·7-s − 0.875·11-s − 1.91i·13-s + 0.105i·17-s + 1.70·19-s − 1.19i·23-s + (−0.109 + 0.994i)25-s + 1.96·29-s + 0.00111·31-s + (−0.200 + 0.179i)35-s + 0.164i·37-s + 0.593·41-s − 0.727i·43-s − 0.492i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.744 + 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.438481809\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.438481809\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.49 + 1.66i)T \) |
| 37 | \( 1 - iT \) |
| good | 7 | \( 1 + 0.712iT - 7T^{2} \) |
| 11 | \( 1 + 2.90T + 11T^{2} \) |
| 13 | \( 1 + 6.88iT - 13T^{2} \) |
| 17 | \( 1 - 0.433iT - 17T^{2} \) |
| 19 | \( 1 - 7.42T + 19T^{2} \) |
| 23 | \( 1 + 5.73iT - 23T^{2} \) |
| 29 | \( 1 - 10.5T + 29T^{2} \) |
| 31 | \( 1 - 0.00620T + 31T^{2} \) |
| 41 | \( 1 - 3.79T + 41T^{2} \) |
| 43 | \( 1 + 4.77iT - 43T^{2} \) |
| 47 | \( 1 + 3.37iT - 47T^{2} \) |
| 53 | \( 1 + 0.266iT - 53T^{2} \) |
| 59 | \( 1 + 4.17T + 59T^{2} \) |
| 61 | \( 1 + 6.75T + 61T^{2} \) |
| 67 | \( 1 - 11.4iT - 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 + 11.1iT - 73T^{2} \) |
| 79 | \( 1 - 4.76T + 79T^{2} \) |
| 83 | \( 1 + 1.61iT - 83T^{2} \) |
| 89 | \( 1 - 7.65T + 89T^{2} \) |
| 97 | \( 1 + 1.08iT - 97T^{2} \) |
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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
−7.79748507519299552331391214848, −7.29131716580287837505176346722, −6.25556061044451111520048847748, −5.30169518066675919081966832052, −5.06786734389432093285985929184, −4.12467105931197981375596456242, −3.18917916204440896141874319303, −2.65084509611681363183268740332, −1.05323234861986662855219559691, −0.45380491288601853889267212810,
1.16291393937371021977356421746, 2.36814404228546786634236733831, 3.03537013820517196158351813171, 3.86228137041078504287404560679, 4.66440754736820740579586046743, 5.35800532920640520613210584567, 6.34639489426310357370293593980, 6.85176476676938703773037193401, 7.66095230459223294596821101214, 7.966423453947314613942025065530
