| L(s) = 1 | + 4.64·3-s + 18.3·5-s − 7·7-s − 5.38·9-s + 39.5·11-s − 64.5·13-s + 85.2·15-s + 109.·17-s + 137.·19-s − 32.5·21-s − 45.2·23-s + 211.·25-s − 150.·27-s + 41.1·29-s + 262.·31-s + 184·33-s − 128.·35-s − 125.·37-s − 299.·39-s − 299.·41-s + 36.9·43-s − 98.7·45-s + 122.·47-s + 49·49-s + 507.·51-s + 20.4·53-s + 725.·55-s + ⋯ |
| L(s) = 1 | + 0.894·3-s + 1.63·5-s − 0.377·7-s − 0.199·9-s + 1.08·11-s − 1.37·13-s + 1.46·15-s + 1.55·17-s + 1.65·19-s − 0.338·21-s − 0.410·23-s + 1.68·25-s − 1.07·27-s + 0.263·29-s + 1.52·31-s + 0.970·33-s − 0.619·35-s − 0.558·37-s − 1.23·39-s − 1.14·41-s + 0.130·43-s − 0.327·45-s + 0.381·47-s + 0.142·49-s + 1.39·51-s + 0.0530·53-s + 1.77·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.534395964\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.534395964\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 3 | \( 1 - 4.64T + 27T^{2} \) |
| 5 | \( 1 - 18.3T + 125T^{2} \) |
| 11 | \( 1 - 39.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 64.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 109.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 137.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 45.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 41.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 262.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 125.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 299.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 36.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 122.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 20.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + 60.8T + 2.05e5T^{2} \) |
| 61 | \( 1 + 791.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.04e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 407.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 562.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 601.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 652.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 898.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 621.T + 9.12e5T^{2} \) |
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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
−10.12403866012600281130442891663, −9.728917805532807568966406538560, −9.180652938648264341081899114033, −8.025819700228228576250898279123, −6.96728386966855304842691162848, −5.91901564828315865149073237603, −5.09299878756849307513215298212, −3.37461920163295047860819015647, −2.53506010118222026204505841790, −1.29925975829022415025363740226,
1.29925975829022415025363740226, 2.53506010118222026204505841790, 3.37461920163295047860819015647, 5.09299878756849307513215298212, 5.91901564828315865149073237603, 6.96728386966855304842691162848, 8.025819700228228576250898279123, 9.180652938648264341081899114033, 9.728917805532807568966406538560, 10.12403866012600281130442891663
