| L(s) = 1 | − 3-s + 2·5-s − 7-s + 9-s − 2·11-s − 4·13-s − 2·15-s + 17-s − 2·19-s + 21-s + 4·23-s − 25-s − 27-s + 2·33-s − 2·35-s + 8·37-s + 4·39-s − 2·41-s − 4·43-s + 2·45-s + 49-s − 51-s + 6·53-s − 4·55-s + 2·57-s − 10·59-s + 6·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s − 1.10·13-s − 0.516·15-s + 0.242·17-s − 0.458·19-s + 0.218·21-s + 0.834·23-s − 1/5·25-s − 0.192·27-s + 0.348·33-s − 0.338·35-s + 1.31·37-s + 0.640·39-s − 0.312·41-s − 0.609·43-s + 0.298·45-s + 1/7·49-s − 0.140·51-s + 0.824·53-s − 0.539·55-s + 0.264·57-s − 1.30·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.82343473365998, −15.07147450239939, −14.84826989962844, −14.08839670418995, −13.40819667564058, −13.13361045180821, −12.52242098320957, −12.03362282259354, −11.39869160036649, −10.69627009334341, −10.30349429473050, −9.666780096074871, −9.405748463977950, −8.587226596547668, −7.785317043739198, −7.323540603664431, −6.535654650588168, −6.168945150838954, −5.365871419751947, −5.042655296495719, −4.304234840589472, −3.379433625958561, −2.555449233346792, −2.050248338829711, −0.9906444742937361, 0,
0.9906444742937361, 2.050248338829711, 2.555449233346792, 3.379433625958561, 4.304234840589472, 5.042655296495719, 5.365871419751947, 6.168945150838954, 6.535654650588168, 7.323540603664431, 7.785317043739198, 8.587226596547668, 9.405748463977950, 9.666780096074871, 10.30349429473050, 10.69627009334341, 11.39869160036649, 12.03362282259354, 12.52242098320957, 13.13361045180821, 13.40819667564058, 14.08839670418995, 14.84826989962844, 15.07147450239939, 15.82343473365998
