| L(s) = 1 | + 5-s − 4·11-s − 2·13-s − 6·17-s − 4·19-s + 8·23-s + 25-s + 6·29-s + 10·37-s + 6·41-s − 43-s + 8·47-s − 7·49-s − 2·53-s − 4·55-s − 4·59-s + 14·61-s − 2·65-s + 4·67-s − 16·71-s + 10·73-s − 8·79-s + 16·83-s − 6·85-s − 6·89-s − 4·95-s + 2·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.20·11-s − 0.554·13-s − 1.45·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s + 1.11·29-s + 1.64·37-s + 0.937·41-s − 0.152·43-s + 1.16·47-s − 49-s − 0.274·53-s − 0.539·55-s − 0.520·59-s + 1.79·61-s − 0.248·65-s + 0.488·67-s − 1.89·71-s + 1.17·73-s − 0.900·79-s + 1.75·83-s − 0.650·85-s − 0.635·89-s − 0.410·95-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.812222288\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.812222288\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.24004638797505, −13.14039524826557, −12.77506950438758, −12.21766556489985, −11.41093352984167, −11.11944542060511, −10.57497250234257, −10.30302889059064, −9.524458468375385, −9.243907767237601, −8.578213567208278, −8.241448921627627, −7.540382711666776, −7.108674031483434, −6.470839681032049, −6.159135475272476, −5.382528718705346, −4.869284683277979, −4.544844030122817, −3.886670362796922, −2.872855102366318, −2.583182341799086, −2.163874194120342, −1.170273559805018, −0.4259551721571167,
0.4259551721571167, 1.170273559805018, 2.163874194120342, 2.583182341799086, 2.872855102366318, 3.886670362796922, 4.544844030122817, 4.869284683277979, 5.382528718705346, 6.159135475272476, 6.470839681032049, 7.108674031483434, 7.540382711666776, 8.241448921627627, 8.578213567208278, 9.243907767237601, 9.524458468375385, 10.30302889059064, 10.57497250234257, 11.11944542060511, 11.41093352984167, 12.21766556489985, 12.77506950438758, 13.14039524826557, 13.24004638797505
