| L(s) = 1 | − 3-s − 2·5-s + 9-s − 4·11-s − 2·13-s + 2·15-s − 2·17-s − 4·19-s − 25-s − 27-s − 29-s + 8·31-s + 4·33-s + 10·37-s + 2·39-s + 6·41-s − 12·43-s − 2·45-s + 8·47-s + 2·51-s − 6·53-s + 8·55-s + 4·57-s + 12·59-s − 10·61-s + 4·65-s + 12·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.516·15-s − 0.485·17-s − 0.917·19-s − 1/5·25-s − 0.192·27-s − 0.185·29-s + 1.43·31-s + 0.696·33-s + 1.64·37-s + 0.320·39-s + 0.937·41-s − 1.82·43-s − 0.298·45-s + 1.16·47-s + 0.280·51-s − 0.824·53-s + 1.07·55-s + 0.529·57-s + 1.56·59-s − 1.28·61-s + 0.496·65-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 272832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9598787249\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9598787249\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−12.75376726101126, −12.34870886186942, −11.63086899047820, −11.54151201338193, −11.04839698205534, −10.39304781481673, −10.19916251818318, −9.686337520561567, −9.005781728882542, −8.489238744604393, −8.024587978005645, −7.618880050431598, −7.275805343925423, −6.576150583032106, −6.158909186235998, −5.685851638489171, −5.000177695237483, −4.504441380779510, −4.354299451744864, −3.557652830907396, −2.944674437594282, −2.372035924014351, −1.886882030196808, −0.8011785468840494, −0.3656201895656679,
0.3656201895656679, 0.8011785468840494, 1.886882030196808, 2.372035924014351, 2.944674437594282, 3.557652830907396, 4.354299451744864, 4.504441380779510, 5.000177695237483, 5.685851638489171, 6.158909186235998, 6.576150583032106, 7.275805343925423, 7.618880050431598, 8.024587978005645, 8.489238744604393, 9.005781728882542, 9.686337520561567, 10.19916251818318, 10.39304781481673, 11.04839698205534, 11.54151201338193, 11.63086899047820, 12.34870886186942, 12.75376726101126
