| L(s) = 1 | − 2·3-s + 4·5-s − 7-s + 9-s − 8·15-s − 2·19-s + 2·21-s − 8·23-s + 11·25-s + 4·27-s − 2·29-s − 4·31-s − 4·35-s + 6·37-s + 2·41-s + 8·43-s + 4·45-s − 4·47-s + 49-s − 10·53-s + 4·57-s + 6·59-s − 4·61-s − 63-s − 12·67-s + 16·69-s + 14·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.78·5-s − 0.377·7-s + 1/3·9-s − 2.06·15-s − 0.458·19-s + 0.436·21-s − 1.66·23-s + 11/5·25-s + 0.769·27-s − 0.371·29-s − 0.718·31-s − 0.676·35-s + 0.986·37-s + 0.312·41-s + 1.21·43-s + 0.596·45-s − 0.583·47-s + 1/7·49-s − 1.37·53-s + 0.529·57-s + 0.781·59-s − 0.512·61-s − 0.125·63-s − 1.46·67-s + 1.92·69-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 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
−16.23068043923322, −16.10280419154453, −15.03083564383292, −14.44283905295230, −13.99202937469052, −13.41699741055303, −12.80434981238659, −12.45588652468938, −11.75881308194996, −11.04608068192679, −10.57916993527701, −10.12861970607718, −9.419428829725499, −9.179219200753906, −8.206657037707064, −7.463513953160994, −6.494777004277076, −6.222458996642793, −5.828675567963224, −5.210299020475793, −4.567431174389515, −3.632878634764248, −2.570517570827447, −2.013768905793029, −1.105631930045298, 0,
1.105631930045298, 2.013768905793029, 2.570517570827447, 3.632878634764248, 4.567431174389515, 5.210299020475793, 5.828675567963224, 6.222458996642793, 6.494777004277076, 7.463513953160994, 8.206657037707064, 9.179219200753906, 9.419428829725499, 10.12861970607718, 10.57916993527701, 11.04608068192679, 11.75881308194996, 12.45588652468938, 12.80434981238659, 13.41699741055303, 13.99202937469052, 14.44283905295230, 15.03083564383292, 16.10280419154453, 16.23068043923322
