| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s + 4·11-s − 2·13-s − 15-s + 4·17-s − 4·19-s − 21-s − 23-s + 25-s + 27-s + 8·29-s − 8·31-s + 4·33-s + 35-s + 2·37-s − 2·39-s + 4·41-s − 8·43-s − 45-s + 49-s + 4·51-s + 4·53-s − 4·55-s − 4·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.258·15-s + 0.970·17-s − 0.917·19-s − 0.218·21-s − 0.208·23-s + 1/5·25-s + 0.192·27-s + 1.48·29-s − 1.43·31-s + 0.696·33-s + 0.169·35-s + 0.328·37-s − 0.320·39-s + 0.624·41-s − 1.21·43-s − 0.149·45-s + 1/7·49-s + 0.560·51-s + 0.549·53-s − 0.539·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 154560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−13.61804874521797, −12.99710590500511, −12.49312111214459, −12.27229810092730, −11.73547745515447, −11.21147840273173, −10.64058287282207, −10.12645372696351, −9.641136564722149, −9.264424974543253, −8.729325441038907, −8.179522309669112, −7.884793402645310, −7.108130676770095, −6.810960408372887, −6.321466589732371, −5.659064385807030, −5.032652018221471, −4.408180244261230, −3.886096655377802, −3.522930731221139, −2.857588370093298, −2.276430170316476, −1.535687637136417, −0.9038802680780476, 0,
0.9038802680780476, 1.535687637136417, 2.276430170316476, 2.857588370093298, 3.522930731221139, 3.886096655377802, 4.408180244261230, 5.032652018221471, 5.659064385807030, 6.321466589732371, 6.810960408372887, 7.108130676770095, 7.884793402645310, 8.179522309669112, 8.729325441038907, 9.264424974543253, 9.641136564722149, 10.12645372696351, 10.64058287282207, 11.21147840273173, 11.73547745515447, 12.27229810092730, 12.49312111214459, 12.99710590500511, 13.61804874521797
