| L(s) = 1 | − 1.34·2-s − 0.184·4-s − 1.34·5-s + 1.22·7-s + 2.94·8-s + 1.81·10-s − 4.59·11-s + 3.59·13-s − 1.65·14-s − 3.59·16-s − 0.305·17-s + 19-s + 0.248·20-s + 6.19·22-s − 4.65·23-s − 3.18·25-s − 4.84·26-s − 0.226·28-s − 1.65·29-s − 7.04·31-s − 1.04·32-s + 0.411·34-s − 1.65·35-s + 0.184·37-s − 1.34·38-s − 3.96·40-s − 3.43·41-s + ⋯ |
| L(s) = 1 | − 0.952·2-s − 0.0923·4-s − 0.602·5-s + 0.463·7-s + 1.04·8-s + 0.574·10-s − 1.38·11-s + 0.997·13-s − 0.441·14-s − 0.899·16-s − 0.0740·17-s + 0.229·19-s + 0.0556·20-s + 1.32·22-s − 0.970·23-s − 0.636·25-s − 0.950·26-s − 0.0428·28-s − 0.306·29-s − 1.26·31-s − 0.184·32-s + 0.0705·34-s − 0.279·35-s + 0.0303·37-s − 0.218·38-s − 0.627·40-s − 0.535·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.34T + 2T^{2} \) |
| 5 | \( 1 + 1.34T + 5T^{2} \) |
| 7 | \( 1 - 1.22T + 7T^{2} \) |
| 11 | \( 1 + 4.59T + 11T^{2} \) |
| 13 | \( 1 - 3.59T + 13T^{2} \) |
| 17 | \( 1 + 0.305T + 17T^{2} \) |
| 23 | \( 1 + 4.65T + 23T^{2} \) |
| 29 | \( 1 + 1.65T + 29T^{2} \) |
| 31 | \( 1 + 7.04T + 31T^{2} \) |
| 37 | \( 1 - 0.184T + 37T^{2} \) |
| 41 | \( 1 + 3.43T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 - 5.58T + 47T^{2} \) |
| 53 | \( 1 - 2.94T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 + 2.40T + 61T^{2} \) |
| 67 | \( 1 - 7.63T + 67T^{2} \) |
| 71 | \( 1 - 1.29T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 - 8.27T + 83T^{2} \) |
| 89 | \( 1 - 7.90T + 89T^{2} \) |
| 97 | \( 1 - 7.00T + 97T^{2} \) |
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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
−10.44819095065112275191029607695, −9.532482103314124346760523125561, −8.471480367138694955314793480696, −8.002457311496096839760859237776, −7.25951652794405745198171058812, −5.75032033858890338540595128429, −4.69781917115460782118619522134, −3.57054299876932205406774443148, −1.79277195841784189019435412536, 0,
1.79277195841784189019435412536, 3.57054299876932205406774443148, 4.69781917115460782118619522134, 5.75032033858890338540595128429, 7.25951652794405745198171058812, 8.002457311496096839760859237776, 8.471480367138694955314793480696, 9.532482103314124346760523125561, 10.44819095065112275191029607695
