| L(s) = 1 | − 0.406·2-s − 0.564·3-s − 1.83·4-s − 2.08·5-s + 0.229·6-s − 4.31·7-s + 1.55·8-s − 2.68·9-s + 0.845·10-s − 1.35·11-s + 1.03·12-s + 1.75·14-s + 1.17·15-s + 3.03·16-s + 3.04·17-s + 1.08·18-s − 4.46·19-s + 3.82·20-s + 2.43·21-s + 0.551·22-s + 5.93·23-s − 0.878·24-s − 0.665·25-s + 3.20·27-s + 7.92·28-s + 2.78·29-s − 0.477·30-s + ⋯ |
| L(s) = 1 | − 0.287·2-s − 0.325·3-s − 0.917·4-s − 0.931·5-s + 0.0935·6-s − 1.63·7-s + 0.550·8-s − 0.893·9-s + 0.267·10-s − 0.409·11-s + 0.298·12-s + 0.468·14-s + 0.303·15-s + 0.759·16-s + 0.738·17-s + 0.256·18-s − 1.02·19-s + 0.854·20-s + 0.531·21-s + 0.117·22-s + 1.23·23-s − 0.179·24-s − 0.133·25-s + 0.616·27-s + 1.49·28-s + 0.516·29-s − 0.0870·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 0.406T + 2T^{2} \) |
| 3 | \( 1 + 0.564T + 3T^{2} \) |
| 5 | \( 1 + 2.08T + 5T^{2} \) |
| 7 | \( 1 + 4.31T + 7T^{2} \) |
| 11 | \( 1 + 1.35T + 11T^{2} \) |
| 17 | \( 1 - 3.04T + 17T^{2} \) |
| 19 | \( 1 + 4.46T + 19T^{2} \) |
| 23 | \( 1 - 5.93T + 23T^{2} \) |
| 29 | \( 1 - 2.78T + 29T^{2} \) |
| 37 | \( 1 + 3.68T + 37T^{2} \) |
| 41 | \( 1 - 0.350T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 + 2.37T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 - 6.18T + 61T^{2} \) |
| 67 | \( 1 - 2.80T + 67T^{2} \) |
| 71 | \( 1 - 11.5T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 - 5.49T + 79T^{2} \) |
| 83 | \( 1 + 6.19T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 + 3.24T + 97T^{2} \) |
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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
−8.073828935585400972141760981554, −7.08492912440689488064753843724, −6.51302466546664907466654155150, −5.55913737670710914860193808050, −5.03250611528880902709581266368, −3.90588253360548993824319600444, −3.47649173954279012870737775652, −2.59747837094786999148903376431, −0.74834287980154469741563706390, 0,
0.74834287980154469741563706390, 2.59747837094786999148903376431, 3.47649173954279012870737775652, 3.90588253360548993824319600444, 5.03250611528880902709581266368, 5.55913737670710914860193808050, 6.51302466546664907466654155150, 7.08492912440689488064753843724, 8.073828935585400972141760981554
