| L(s) = 1 | + 3-s + 9-s − 6·11-s − 2·13-s − 6·17-s − 4·19-s + 8·23-s + 27-s + 8·31-s − 6·33-s − 2·37-s − 2·39-s + 6·41-s − 4·43-s − 4·47-s − 6·51-s − 6·53-s − 4·57-s + 6·59-s + 6·61-s + 8·69-s − 4·71-s + 12·73-s − 8·79-s + 81-s + 12·83-s − 14·89-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.80·11-s − 0.554·13-s − 1.45·17-s − 0.917·19-s + 1.66·23-s + 0.192·27-s + 1.43·31-s − 1.04·33-s − 0.328·37-s − 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.583·47-s − 0.840·51-s − 0.824·53-s − 0.529·57-s + 0.781·59-s + 0.768·61-s + 0.963·69-s − 0.474·71-s + 1.40·73-s − 0.900·79-s + 1/9·81-s + 1.31·83-s − 1.48·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.65205465079139, −13.26636094734453, −13.02802373763937, −12.56701826211166, −11.99194269421670, −11.18724889866668, −10.88594646697551, −10.53631074887778, −9.780766033301583, −9.579185983770314, −8.723310344786123, −8.450161279264693, −8.029190323348728, −7.385194982629242, −6.884542600051537, −6.502033486079508, −5.717817115751901, −5.028295413332631, −4.731399312510904, −4.220894345950733, −3.341698285087256, −2.714778206101973, −2.460034863100088, −1.813079021647617, −0.7646022623729650, 0,
0.7646022623729650, 1.813079021647617, 2.460034863100088, 2.714778206101973, 3.341698285087256, 4.220894345950733, 4.731399312510904, 5.028295413332631, 5.717817115751901, 6.502033486079508, 6.884542600051537, 7.385194982629242, 8.029190323348728, 8.450161279264693, 8.723310344786123, 9.579185983770314, 9.780766033301583, 10.53631074887778, 10.88594646697551, 11.18724889866668, 11.99194269421670, 12.56701826211166, 13.02802373763937, 13.26636094734453, 13.65205465079139
