| L(s) = 1 | − 1.22·2-s − 0.451·3-s − 0.488·4-s + 5-s + 0.554·6-s − 2.48·7-s + 3.05·8-s − 2.79·9-s − 1.22·10-s − 11-s + 0.220·12-s + 0.572·13-s + 3.05·14-s − 0.451·15-s − 2.78·16-s − 4.98·17-s + 3.43·18-s − 0.991·19-s − 0.488·20-s + 1.12·21-s + 1.22·22-s + 7.38·23-s − 1.37·24-s + 25-s − 0.704·26-s + 2.61·27-s + 1.21·28-s + ⋯ |
| L(s) = 1 | − 0.869·2-s − 0.260·3-s − 0.244·4-s + 0.447·5-s + 0.226·6-s − 0.940·7-s + 1.08·8-s − 0.932·9-s − 0.388·10-s − 0.301·11-s + 0.0635·12-s + 0.158·13-s + 0.817·14-s − 0.116·15-s − 0.696·16-s − 1.20·17-s + 0.810·18-s − 0.227·19-s − 0.109·20-s + 0.244·21-s + 0.262·22-s + 1.54·23-s − 0.281·24-s + 0.200·25-s − 0.138·26-s + 0.503·27-s + 0.229·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 + 1.22T + 2T^{2} \) |
| 3 | \( 1 + 0.451T + 3T^{2} \) |
| 7 | \( 1 + 2.48T + 7T^{2} \) |
| 13 | \( 1 - 0.572T + 13T^{2} \) |
| 17 | \( 1 + 4.98T + 17T^{2} \) |
| 19 | \( 1 + 0.991T + 19T^{2} \) |
| 23 | \( 1 - 7.38T + 23T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 - 3.51T + 31T^{2} \) |
| 37 | \( 1 + 5.05T + 37T^{2} \) |
| 41 | \( 1 - 4.09T + 41T^{2} \) |
| 43 | \( 1 - 1.31T + 43T^{2} \) |
| 47 | \( 1 - 4.64T + 47T^{2} \) |
| 53 | \( 1 - 6.59T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 5.18T + 61T^{2} \) |
| 67 | \( 1 + 4.75T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 79 | \( 1 - 6.70T + 79T^{2} \) |
| 83 | \( 1 + 0.880T + 83T^{2} \) |
| 89 | \( 1 + 0.942T + 89T^{2} \) |
| 97 | \( 1 + 4.05T + 97T^{2} \) |
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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
−8.404337342409068399101176938589, −7.38497687254498498069893955446, −6.61267811788405896177960887893, −6.06378740012536711324004159906, −5.04587858105558722923394443623, −4.45153421172282161181380241067, −3.16616089083624028013228858476, −2.45070445353154878859989096224, −1.04249088626989250781039409564, 0,
1.04249088626989250781039409564, 2.45070445353154878859989096224, 3.16616089083624028013228858476, 4.45153421172282161181380241067, 5.04587858105558722923394443623, 6.06378740012536711324004159906, 6.61267811788405896177960887893, 7.38497687254498498069893955446, 8.404337342409068399101176938589
