| 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 + 6·29-s + 8·31-s + 4·33-s − 6·37-s + 2·39-s − 6·41-s − 4·43-s + 2·45-s − 7·49-s + 2·51-s + 2·53-s − 8·55-s − 4·57-s + 4·59-s + 2·61-s − 4·65-s + 4·67-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 + 1.11·29-s + 1.43·31-s + 0.696·33-s − 0.986·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s + 0.298·45-s − 49-s + 0.280·51-s + 0.274·53-s − 1.07·55-s − 0.529·57-s + 0.520·59-s + 0.256·61-s − 0.496·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.58593172334788, −16.00601619749672, −15.51806779585239, −15.06094949899946, −14.05870319008221, −13.74977549082251, −13.26560976153551, −12.59945965060409, −12.00035150878379, −11.52261679664918, −10.68353274937486, −10.22172520089540, −9.821544784635579, −9.217247137528152, −8.203132085402416, −7.932741438944089, −6.810918917143638, −6.618948213116475, −5.686930301735402, −5.086374372400919, −4.829312432515918, −3.665115293059204, −2.741283731455776, −2.155783072374226, −1.138507899479109, 0,
1.138507899479109, 2.155783072374226, 2.741283731455776, 3.665115293059204, 4.829312432515918, 5.086374372400919, 5.686930301735402, 6.618948213116475, 6.810918917143638, 7.932741438944089, 8.203132085402416, 9.217247137528152, 9.821544784635579, 10.22172520089540, 10.68353274937486, 11.52261679664918, 12.00035150878379, 12.59945965060409, 13.26560976153551, 13.74977549082251, 14.05870319008221, 15.06094949899946, 15.51806779585239, 16.00601619749672, 16.58593172334788
