| L(s) = 1 | − 1.85·2-s + 3.31·3-s + 1.45·4-s + 0.943·5-s − 6.15·6-s − 1.28·7-s + 1.01·8-s + 7.97·9-s − 1.75·10-s − 11-s + 4.82·12-s − 3.54·13-s + 2.37·14-s + 3.12·15-s − 4.79·16-s + 17-s − 14.8·18-s − 3.35·19-s + 1.37·20-s − 4.24·21-s + 1.85·22-s + 3.29·23-s + 3.35·24-s − 4.11·25-s + 6.58·26-s + 16.4·27-s − 1.86·28-s + ⋯ |
| L(s) = 1 | − 1.31·2-s + 1.91·3-s + 0.727·4-s + 0.421·5-s − 2.51·6-s − 0.483·7-s + 0.357·8-s + 2.65·9-s − 0.554·10-s − 0.301·11-s + 1.39·12-s − 0.982·13-s + 0.636·14-s + 0.806·15-s − 1.19·16-s + 0.242·17-s − 3.49·18-s − 0.770·19-s + 0.306·20-s − 0.925·21-s + 0.396·22-s + 0.687·23-s + 0.684·24-s − 0.822·25-s + 1.29·26-s + 3.17·27-s − 0.352·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 1.85T + 2T^{2} \) |
| 3 | \( 1 - 3.31T + 3T^{2} \) |
| 5 | \( 1 - 0.943T + 5T^{2} \) |
| 7 | \( 1 + 1.28T + 7T^{2} \) |
| 13 | \( 1 + 3.54T + 13T^{2} \) |
| 19 | \( 1 + 3.35T + 19T^{2} \) |
| 23 | \( 1 - 3.29T + 23T^{2} \) |
| 29 | \( 1 + 6.51T + 29T^{2} \) |
| 31 | \( 1 - 0.322T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 + 8.39T + 41T^{2} \) |
| 47 | \( 1 - 6.51T + 47T^{2} \) |
| 53 | \( 1 - 3.17T + 53T^{2} \) |
| 59 | \( 1 - 1.42T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 + 4.14T + 71T^{2} \) |
| 73 | \( 1 + 6.21T + 73T^{2} \) |
| 79 | \( 1 + 16.6T + 79T^{2} \) |
| 83 | \( 1 + 0.588T + 83T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 + 17.6T + 97T^{2} \) |
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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
−7.63513264605734314155573321541, −7.23208137739616314175288357793, −6.63352827741489823921858601014, −5.34988064538946501867722284508, −4.44162075961870778703119066769, −3.63636181920291794210359623383, −2.81120145505754332305213287754, −2.08264651635893479697538488730, −1.52953842398780976924654486174, 0,
1.52953842398780976924654486174, 2.08264651635893479697538488730, 2.81120145505754332305213287754, 3.63636181920291794210359623383, 4.44162075961870778703119066769, 5.34988064538946501867722284508, 6.63352827741489823921858601014, 7.23208137739616314175288357793, 7.63513264605734314155573321541
