| L(s) = 1 | + 3-s + 5-s − 2·7-s + 9-s + 11-s − 2·13-s + 15-s + 2·19-s − 2·21-s + 25-s + 27-s − 8·31-s + 33-s − 2·35-s − 2·37-s − 2·39-s + 2·43-s + 45-s − 3·49-s − 6·53-s + 55-s + 2·57-s − 12·59-s − 2·61-s − 2·63-s − 2·65-s − 4·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s + 0.258·15-s + 0.458·19-s − 0.436·21-s + 1/5·25-s + 0.192·27-s − 1.43·31-s + 0.174·33-s − 0.338·35-s − 0.328·37-s − 0.320·39-s + 0.304·43-s + 0.149·45-s − 3/7·49-s − 0.824·53-s + 0.134·55-s + 0.264·57-s − 1.56·59-s − 0.256·61-s − 0.251·63-s − 0.248·65-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 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
−16.75766005174902, −16.32619084212038, −15.64995058689201, −15.14383641172566, −14.42746555289729, −14.06713785531499, −13.46260920620244, −12.74361094733854, −12.51315523808591, −11.68112963355511, −10.95641562420264, −10.29789562563396, −9.670703707952690, −9.264644051640369, −8.758457023862599, −7.815628499316884, −7.334126845248906, −6.619687799712750, −6.010120061636994, −5.233796240597086, −4.492687793847700, −3.575555413711445, −3.057167068897381, −2.194682463362011, −1.367493615008676, 0,
1.367493615008676, 2.194682463362011, 3.057167068897381, 3.575555413711445, 4.492687793847700, 5.233796240597086, 6.010120061636994, 6.619687799712750, 7.334126845248906, 7.815628499316884, 8.758457023862599, 9.264644051640369, 9.670703707952690, 10.29789562563396, 10.95641562420264, 11.68112963355511, 12.51315523808591, 12.74361094733854, 13.46260920620244, 14.06713785531499, 14.42746555289729, 15.14383641172566, 15.64995058689201, 16.32619084212038, 16.75766005174902
