| L(s) = 1 | − 5-s − 3·9-s − 4·11-s − 6·13-s − 2·17-s + 25-s − 6·29-s − 8·31-s + 10·37-s − 2·41-s − 4·43-s + 3·45-s − 8·47-s + 2·53-s + 4·55-s − 8·59-s − 14·61-s + 6·65-s + 12·67-s − 16·71-s − 2·73-s − 8·79-s + 9·81-s + 8·83-s + 2·85-s − 10·89-s − 2·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 9-s − 1.20·11-s − 1.66·13-s − 0.485·17-s + 1/5·25-s − 1.11·29-s − 1.43·31-s + 1.64·37-s − 0.312·41-s − 0.609·43-s + 0.447·45-s − 1.16·47-s + 0.274·53-s + 0.539·55-s − 1.04·59-s − 1.79·61-s + 0.744·65-s + 1.46·67-s − 1.89·71-s − 0.234·73-s − 0.900·79-s + 81-s + 0.878·83-s + 0.216·85-s − 1.05·89-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−16.56739417988620, −16.10570336370448, −15.22555519722006, −14.90411463684232, −14.55600118312933, −13.75999366610562, −13.09290548407192, −12.73998338162403, −12.02375700962658, −11.50718729619303, −10.94072379012947, −10.47802286686198, −9.582269129704764, −9.292696971976882, −8.424906545652173, −7.820828551097045, −7.485835514320124, −6.756087765363150, −5.870924518987700, −5.296648544264432, −4.792144891504565, −4.002244358002195, −2.995893500312555, −2.635901560358558, −1.745060466171496, 0, 0,
1.745060466171496, 2.635901560358558, 2.995893500312555, 4.002244358002195, 4.792144891504565, 5.296648544264432, 5.870924518987700, 6.756087765363150, 7.485835514320124, 7.820828551097045, 8.424906545652173, 9.292696971976882, 9.582269129704764, 10.47802286686198, 10.94072379012947, 11.50718729619303, 12.02375700962658, 12.73998338162403, 13.09290548407192, 13.75999366610562, 14.55600118312933, 14.90411463684232, 15.22555519722006, 16.10570336370448, 16.56739417988620
