| L(s) = 1 | − 3-s + 9-s + 11-s − 13-s − 2·17-s + 5·19-s − 6·23-s − 27-s − 2·29-s − 3·31-s − 33-s − 10·37-s + 39-s + 6·41-s + 3·43-s − 4·47-s + 2·51-s − 10·53-s − 5·57-s − 4·59-s − 5·61-s + 5·67-s + 6·69-s + 14·73-s + 4·79-s + 81-s + 6·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.301·11-s − 0.277·13-s − 0.485·17-s + 1.14·19-s − 1.25·23-s − 0.192·27-s − 0.371·29-s − 0.538·31-s − 0.174·33-s − 1.64·37-s + 0.160·39-s + 0.937·41-s + 0.457·43-s − 0.583·47-s + 0.280·51-s − 1.37·53-s − 0.662·57-s − 0.520·59-s − 0.640·61-s + 0.610·67-s + 0.722·69-s + 1.63·73-s + 0.450·79-s + 1/9·81-s + 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 15 T + p T^{2} \) |
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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
−12.70333501582027, −12.27817459312769, −12.04233237144491, −11.45125295435367, −11.07036694424650, −10.63816680143610, −10.17418832978547, −9.526743997796427, −9.363641312302305, −8.851176795850176, −8.023935287093029, −7.821232413786819, −7.302879840857776, −6.589193463430780, −6.484816668520778, −5.722883467692467, −5.338505350138762, −4.905303385755982, −4.280098653304470, −3.770954326422107, −3.314985456556251, −2.596973482884796, −1.895831514890889, −1.502130112138322, −0.6519338067677989, 0,
0.6519338067677989, 1.502130112138322, 1.895831514890889, 2.596973482884796, 3.314985456556251, 3.770954326422107, 4.280098653304470, 4.905303385755982, 5.338505350138762, 5.722883467692467, 6.484816668520778, 6.589193463430780, 7.302879840857776, 7.821232413786819, 8.023935287093029, 8.851176795850176, 9.363641312302305, 9.526743997796427, 10.17418832978547, 10.63816680143610, 11.07036694424650, 11.45125295435367, 12.04233237144491, 12.27817459312769, 12.70333501582027
