| L(s) = 1 | − 2·2-s + 2·4-s + 5-s + 2·7-s − 2·10-s + 11-s − 4·14-s − 4·16-s + 2·17-s + 2·20-s − 2·22-s + 23-s − 4·25-s + 4·28-s − 7·31-s + 8·32-s − 4·34-s + 2·35-s − 3·37-s − 8·41-s − 6·43-s + 2·44-s − 2·46-s + 8·47-s − 3·49-s + 8·50-s + 6·53-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.447·5-s + 0.755·7-s − 0.632·10-s + 0.301·11-s − 1.06·14-s − 16-s + 0.485·17-s + 0.447·20-s − 0.426·22-s + 0.208·23-s − 4/5·25-s + 0.755·28-s − 1.25·31-s + 1.41·32-s − 0.685·34-s + 0.338·35-s − 0.493·37-s − 1.24·41-s − 0.914·43-s + 0.301·44-s − 0.294·46-s + 1.16·47-s − 3/7·49-s + 1.13·50-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−16.17119370868094, −16.04017694829068, −15.01342045433365, −14.70900274278920, −13.97462389780472, −13.47875339254741, −12.90405649079962, −11.93572730281821, −11.64687795594958, −10.94157749038626, −10.43988364020299, −9.869398335815727, −9.439852380046632, −8.702514307388600, −8.371551985173552, −7.710471249569395, −7.101525498178462, −6.616757870936443, −5.588438602150575, −5.198220839409890, −4.246092921211587, −3.523660296163367, −2.379361287488802, −1.755120387667789, −1.133166789159409, 0,
1.133166789159409, 1.755120387667789, 2.379361287488802, 3.523660296163367, 4.246092921211587, 5.198220839409890, 5.588438602150575, 6.616757870936443, 7.101525498178462, 7.710471249569395, 8.371551985173552, 8.702514307388600, 9.439852380046632, 9.869398335815727, 10.43988364020299, 10.94157749038626, 11.64687795594958, 11.93572730281821, 12.90405649079962, 13.47875339254741, 13.97462389780472, 14.70900274278920, 15.01342045433365, 16.04017694829068, 16.17119370868094
