| L(s) = 1 | − 2.28·2-s + 0.276·3-s + 3.23·4-s − 5-s − 0.631·6-s + 7-s − 2.83·8-s − 2.92·9-s + 2.28·10-s + 2.68·11-s + 0.894·12-s − 6.13·13-s − 2.28·14-s − 0.276·15-s + 0.0155·16-s + 0.623·17-s + 6.69·18-s + 1.16·19-s − 3.23·20-s + 0.276·21-s − 6.13·22-s + 4.46·23-s − 0.783·24-s + 25-s + 14.0·26-s − 1.63·27-s + 3.23·28-s + ⋯ |
| L(s) = 1 | − 1.61·2-s + 0.159·3-s + 1.61·4-s − 0.447·5-s − 0.257·6-s + 0.377·7-s − 1.00·8-s − 0.974·9-s + 0.723·10-s + 0.808·11-s + 0.258·12-s − 1.70·13-s − 0.611·14-s − 0.0712·15-s + 0.00388·16-s + 0.151·17-s + 1.57·18-s + 0.267·19-s − 0.724·20-s + 0.0602·21-s − 1.30·22-s + 0.931·23-s − 0.159·24-s + 0.200·25-s + 2.75·26-s − 0.314·27-s + 0.612·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 + 2.28T + 2T^{2} \) |
| 3 | \( 1 - 0.276T + 3T^{2} \) |
| 11 | \( 1 - 2.68T + 11T^{2} \) |
| 13 | \( 1 + 6.13T + 13T^{2} \) |
| 17 | \( 1 - 0.623T + 17T^{2} \) |
| 19 | \( 1 - 1.16T + 19T^{2} \) |
| 23 | \( 1 - 4.46T + 23T^{2} \) |
| 29 | \( 1 + 7.00T + 29T^{2} \) |
| 31 | \( 1 - 3.39T + 31T^{2} \) |
| 37 | \( 1 - 2.25T + 37T^{2} \) |
| 41 | \( 1 + 3.93T + 41T^{2} \) |
| 43 | \( 1 - 2.63T + 43T^{2} \) |
| 47 | \( 1 - 2.10T + 47T^{2} \) |
| 53 | \( 1 + 14.3T + 53T^{2} \) |
| 59 | \( 1 + 1.12T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 - 1.06T + 73T^{2} \) |
| 79 | \( 1 + 5.44T + 79T^{2} \) |
| 83 | \( 1 + 4.31T + 83T^{2} \) |
| 89 | \( 1 - 17.7T + 89T^{2} \) |
| 97 | \( 1 + 3.89T + 97T^{2} \) |
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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
−7.77579688033354429404606392890, −7.04829554044452958967210176885, −6.53750397725154415723823056794, −5.41687274399304604888616099573, −4.77922435223278913487098782348, −3.70865718667317052923315766940, −2.74259190405116571797833517352, −2.07185272974031460324275217854, −0.980147482577540975250205121262, 0,
0.980147482577540975250205121262, 2.07185272974031460324275217854, 2.74259190405116571797833517352, 3.70865718667317052923315766940, 4.77922435223278913487098782348, 5.41687274399304604888616099573, 6.53750397725154415723823056794, 7.04829554044452958967210176885, 7.77579688033354429404606392890
