| L(s) = 1 | − 3-s − 5-s + 9-s − 4·11-s + 2·13-s + 15-s − 2·17-s − 4·19-s + 8·23-s + 25-s − 27-s + 6·29-s − 8·31-s + 4·33-s − 2·37-s − 2·39-s − 2·41-s + 12·43-s − 45-s − 8·47-s + 2·51-s + 6·53-s + 4·55-s + 4·57-s + 4·59-s + 2·61-s − 2·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.258·15-s − 0.485·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.696·33-s − 0.328·37-s − 0.320·39-s − 0.312·41-s + 1.82·43-s − 0.149·45-s − 1.16·47-s + 0.280·51-s + 0.824·53-s + 0.539·55-s + 0.529·57-s + 0.520·59-s + 0.256·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−16.56416590205003, −16.13036049918681, −15.64859306037773, −15.03412433507701, −14.64828240963199, −13.62978807289840, −13.21258220554690, −12.67307897255452, −12.21845894440993, −11.34461631819386, −10.81322654836839, −10.68106272460413, −9.818386622106549, −8.921718148370296, −8.569355627683011, −7.749153319981662, −7.162863989668186, −6.565305113880707, −5.821385294005866, −5.117941838895465, −4.603171695422540, −3.787136929344603, −2.956646741828617, −2.142948492207969, −0.9887039806302786, 0,
0.9887039806302786, 2.142948492207969, 2.956646741828617, 3.787136929344603, 4.603171695422540, 5.117941838895465, 5.821385294005866, 6.565305113880707, 7.162863989668186, 7.749153319981662, 8.569355627683011, 8.921718148370296, 9.818386622106549, 10.68106272460413, 10.81322654836839, 11.34461631819386, 12.21845894440993, 12.67307897255452, 13.21258220554690, 13.62978807289840, 14.64828240963199, 15.03412433507701, 15.64859306037773, 16.13036049918681, 16.56416590205003
