| L(s) = 1 | − 0.669·3-s − 4.30·5-s + 7-s − 2.55·9-s − 0.669·11-s + 13-s + 2.88·15-s + 6.22·17-s + 6.30·19-s − 0.669·21-s − 5.63·23-s + 13.5·25-s + 3.71·27-s − 1.42·29-s + 4.09·31-s + 0.448·33-s − 4.30·35-s + 2.66·37-s − 0.669·39-s + 7.05·41-s − 10.5·43-s + 10.9·45-s + 4.09·47-s + 49-s − 4.16·51-s + 5.42·53-s + 2.88·55-s + ⋯ |
| L(s) = 1 | − 0.386·3-s − 1.92·5-s + 0.377·7-s − 0.850·9-s − 0.201·11-s + 0.277·13-s + 0.744·15-s + 1.50·17-s + 1.44·19-s − 0.146·21-s − 1.17·23-s + 2.70·25-s + 0.715·27-s − 0.264·29-s + 0.734·31-s + 0.0780·33-s − 0.727·35-s + 0.438·37-s − 0.107·39-s + 1.10·41-s − 1.60·43-s + 1.63·45-s + 0.596·47-s + 0.142·49-s − 0.583·51-s + 0.744·53-s + 0.388·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8440568621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8440568621\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 0.669T + 3T^{2} \) |
| 5 | \( 1 + 4.30T + 5T^{2} \) |
| 11 | \( 1 + 0.669T + 11T^{2} \) |
| 17 | \( 1 - 6.22T + 17T^{2} \) |
| 19 | \( 1 - 6.30T + 19T^{2} \) |
| 23 | \( 1 + 5.63T + 23T^{2} \) |
| 29 | \( 1 + 1.42T + 29T^{2} \) |
| 31 | \( 1 - 4.09T + 31T^{2} \) |
| 37 | \( 1 - 2.66T + 37T^{2} \) |
| 41 | \( 1 - 7.05T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 - 4.09T + 47T^{2} \) |
| 53 | \( 1 - 5.42T + 53T^{2} \) |
| 59 | \( 1 + 7.26T + 59T^{2} \) |
| 61 | \( 1 - 2.28T + 61T^{2} \) |
| 67 | \( 1 - 2.21T + 67T^{2} \) |
| 71 | \( 1 - 5.56T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 - 0.312T + 79T^{2} \) |
| 83 | \( 1 + 3.03T + 83T^{2} \) |
| 89 | \( 1 - 4.12T + 89T^{2} \) |
| 97 | \( 1 - 7.43T + 97T^{2} \) |
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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
−10.66619426784494216400100501958, −9.556171465237468391745063741877, −8.222841813871199306655142183012, −8.013350326109376603180727465829, −7.12266720090285200796781230484, −5.81637975537818377637800513937, −4.94655443070003583486360350995, −3.82919788100451443642320161129, −3.03435789498427288103803756478, −0.78426293209142152061249036003,
0.78426293209142152061249036003, 3.03435789498427288103803756478, 3.82919788100451443642320161129, 4.94655443070003583486360350995, 5.81637975537818377637800513937, 7.12266720090285200796781230484, 8.013350326109376603180727465829, 8.222841813871199306655142183012, 9.556171465237468391745063741877, 10.66619426784494216400100501958
