| L(s) = 1 | + 2-s − 0.749·3-s + 4-s − 0.301·5-s − 0.749·6-s − 4.79·7-s + 8-s − 2.43·9-s − 0.301·10-s − 1.51·11-s − 0.749·12-s − 4.53·13-s − 4.79·14-s + 0.226·15-s + 16-s − 1.24·17-s − 2.43·18-s − 2.66·19-s − 0.301·20-s + 3.59·21-s − 1.51·22-s − 0.816·23-s − 0.749·24-s − 4.90·25-s − 4.53·26-s + 4.07·27-s − 4.79·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.432·3-s + 0.5·4-s − 0.134·5-s − 0.306·6-s − 1.81·7-s + 0.353·8-s − 0.812·9-s − 0.0953·10-s − 0.456·11-s − 0.216·12-s − 1.25·13-s − 1.28·14-s + 0.0584·15-s + 0.250·16-s − 0.302·17-s − 0.574·18-s − 0.612·19-s − 0.0674·20-s + 0.784·21-s − 0.323·22-s − 0.170·23-s − 0.153·24-s − 0.981·25-s − 0.889·26-s + 0.784·27-s − 0.906·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6790435140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6790435140\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 + 0.749T + 3T^{2} \) |
| 5 | \( 1 + 0.301T + 5T^{2} \) |
| 7 | \( 1 + 4.79T + 7T^{2} \) |
| 11 | \( 1 + 1.51T + 11T^{2} \) |
| 13 | \( 1 + 4.53T + 13T^{2} \) |
| 17 | \( 1 + 1.24T + 17T^{2} \) |
| 19 | \( 1 + 2.66T + 19T^{2} \) |
| 23 | \( 1 + 0.816T + 23T^{2} \) |
| 29 | \( 1 + 8.49T + 29T^{2} \) |
| 37 | \( 1 - 6.79T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 - 0.557T + 43T^{2} \) |
| 47 | \( 1 - 0.322T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 + 2.47T + 59T^{2} \) |
| 61 | \( 1 + 8.23T + 61T^{2} \) |
| 67 | \( 1 + 1.07T + 67T^{2} \) |
| 71 | \( 1 - 2.34T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 2.67T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 + 3.56T + 89T^{2} \) |
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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
−7.73423814146657579958751917965, −7.32018059188038039259355583172, −6.35375920536454455208629803811, −5.99518572417610186535759304227, −5.36796667049874220861958218559, −4.39237908989561737492223926856, −3.68738738791685353657851285616, −2.77122741767824317866034460961, −2.31820317124364646765313168956, −0.36093316834464114957765858888,
0.36093316834464114957765858888, 2.31820317124364646765313168956, 2.77122741767824317866034460961, 3.68738738791685353657851285616, 4.39237908989561737492223926856, 5.36796667049874220861958218559, 5.99518572417610186535759304227, 6.35375920536454455208629803811, 7.32018059188038039259355583172, 7.73423814146657579958751917965
