| L(s) = 1 | − 78.0·5-s + 49·7-s − 746.·11-s + 9.75·13-s − 1.75e3·17-s + 603.·19-s − 3.17e3·23-s + 2.96e3·25-s − 3.22e3·29-s − 7.88e3·31-s − 3.82e3·35-s + 1.23e4·37-s − 3.69e3·41-s − 1.69e4·43-s + 658.·47-s + 2.40e3·49-s − 2.78e4·53-s + 5.82e4·55-s + 7.45e3·59-s − 4.37e4·61-s − 761.·65-s + 2.86e4·67-s − 9.24e3·71-s − 2.98e4·73-s − 3.65e4·77-s − 3.29e4·79-s + 4.05e4·83-s + ⋯ |
| L(s) = 1 | − 1.39·5-s + 0.377·7-s − 1.85·11-s + 0.0160·13-s − 1.47·17-s + 0.383·19-s − 1.25·23-s + 0.949·25-s − 0.712·29-s − 1.47·31-s − 0.527·35-s + 1.48·37-s − 0.343·41-s − 1.39·43-s + 0.0434·47-s + 0.142·49-s − 1.36·53-s + 2.59·55-s + 0.278·59-s − 1.50·61-s − 0.0223·65-s + 0.779·67-s − 0.217·71-s − 0.654·73-s − 0.703·77-s − 0.593·79-s + 0.645·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.09865578949\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09865578949\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 49T \) |
| good | 5 | \( 1 + 78.0T + 3.12e3T^{2} \) |
| 11 | \( 1 + 746.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 9.75T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.75e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 603.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.17e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.22e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.88e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.23e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.69e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.69e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 658.T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.78e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 7.45e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.37e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.86e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 9.24e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.98e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.29e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.05e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.13e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.10e5T + 8.58e9T^{2} \) |
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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
−9.110013411779934129236227630656, −7.991562951016727310602571774333, −7.902885696039227995944884136849, −6.92497659327963682156438025707, −5.68361873837266505510381473814, −4.76886433875346986183380356998, −4.01237752265643661341368729524, −2.95253357705356951539059783139, −1.88676019199566580932694813641, −0.12879740626776010187684498022,
0.12879740626776010187684498022, 1.88676019199566580932694813641, 2.95253357705356951539059783139, 4.01237752265643661341368729524, 4.76886433875346986183380356998, 5.68361873837266505510381473814, 6.92497659327963682156438025707, 7.902885696039227995944884136849, 7.991562951016727310602571774333, 9.110013411779934129236227630656
