| L(s) = 1 | + 7.52·3-s − 72.6·5-s + 49·7-s − 186.·9-s + 30.4·11-s + 1.14e3·13-s − 546.·15-s − 514.·17-s − 2.31e3·19-s + 368.·21-s + 409.·23-s + 2.15e3·25-s − 3.23e3·27-s + 1.69e3·29-s − 7.03e3·31-s + 229.·33-s − 3.55e3·35-s − 3.93e3·37-s + 8.61e3·39-s − 9.23e3·41-s + 2.13e4·43-s + 1.35e4·45-s + 9.94e3·47-s + 2.40e3·49-s − 3.86e3·51-s + 3.45e4·53-s − 2.21e3·55-s + ⋯ |
| L(s) = 1 | + 0.482·3-s − 1.29·5-s + 0.377·7-s − 0.767·9-s + 0.0758·11-s + 1.88·13-s − 0.626·15-s − 0.431·17-s − 1.47·19-s + 0.182·21-s + 0.161·23-s + 0.688·25-s − 0.852·27-s + 0.373·29-s − 1.31·31-s + 0.0366·33-s − 0.491·35-s − 0.472·37-s + 0.907·39-s − 0.858·41-s + 1.76·43-s + 0.996·45-s + 0.656·47-s + 0.142·49-s − 0.208·51-s + 1.69·53-s − 0.0985·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.659210319\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.659210319\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 49T \) |
| good | 3 | \( 1 - 7.52T + 243T^{2} \) |
| 5 | \( 1 + 72.6T + 3.12e3T^{2} \) |
| 11 | \( 1 - 30.4T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.14e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 514.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.31e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 409.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.69e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.03e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.93e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 9.23e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.13e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 9.94e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.45e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.63e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.38e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.92e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.95e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.38e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 622.T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.72e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.69e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.54e4T + 8.58e9T^{2} \) |
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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
−10.63675263792136542678834791261, −8.824995112815675572292406097408, −8.642337907496546300923941082455, −7.77779566838458231856904189671, −6.67010917744283595582027193985, −5.57059697311654687593307868106, −4.09292246852151320124879321938, −3.61304861473844111273184107664, −2.19327241555041831319481622945, −0.63917137898367479077913521140,
0.63917137898367479077913521140, 2.19327241555041831319481622945, 3.61304861473844111273184107664, 4.09292246852151320124879321938, 5.57059697311654687593307868106, 6.67010917744283595582027193985, 7.77779566838458231856904189671, 8.642337907496546300923941082455, 8.824995112815675572292406097408, 10.63675263792136542678834791261
