| L(s) = 1 | + 87.0·3-s + 125·5-s + 741.·7-s + 5.38e3·9-s − 6.06e3·11-s − 4.23e3·13-s + 1.08e4·15-s + 1.84e4·17-s − 2.80e4·19-s + 6.45e4·21-s + 1.04e5·23-s + 1.56e4·25-s + 2.78e5·27-s − 2.04e5·29-s + 5.10e3·31-s − 5.27e5·33-s + 9.26e4·35-s + 1.86e5·37-s − 3.68e5·39-s − 6.18e5·41-s − 3.29e5·43-s + 6.73e5·45-s + 5.62e4·47-s − 2.74e5·49-s + 1.60e6·51-s + 1.43e5·53-s − 7.58e5·55-s + ⋯ |
| L(s) = 1 | + 1.86·3-s + 0.447·5-s + 0.816·7-s + 2.46·9-s − 1.37·11-s − 0.534·13-s + 0.832·15-s + 0.912·17-s − 0.938·19-s + 1.52·21-s + 1.79·23-s + 0.199·25-s + 2.72·27-s − 1.55·29-s + 0.0308·31-s − 2.55·33-s + 0.365·35-s + 0.604·37-s − 0.994·39-s − 1.40·41-s − 0.631·43-s + 1.10·45-s + 0.0789·47-s − 0.332·49-s + 1.69·51-s + 0.132·53-s − 0.614·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.554762667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.554762667\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 125T \) |
| good | 3 | \( 1 - 87.0T + 2.18e3T^{2} \) |
| 7 | \( 1 - 741.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 6.06e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 4.23e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.84e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.80e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.04e5T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.04e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 5.10e3T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.86e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.18e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.29e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.62e4T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.43e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 3.09e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 7.01e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.15e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 5.03e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.64e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.67e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.64e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.11e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 4.05e6T + 8.07e13T^{2} \) |
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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
−14.76430445870802188522027398100, −13.56042599780776629641516306583, −12.78265179266623581806616031879, −10.64709360972463975018199306595, −9.458095915650258549112104793374, −8.285004557704822512179849431553, −7.39815046861630954670907124922, −4.92101178388324745186976307624, −3.06153547827333475651716533639, −1.83513512494041941225325762419,
1.83513512494041941225325762419, 3.06153547827333475651716533639, 4.92101178388324745186976307624, 7.39815046861630954670907124922, 8.285004557704822512179849431553, 9.458095915650258549112104793374, 10.64709360972463975018199306595, 12.78265179266623581806616031879, 13.56042599780776629641516306583, 14.76430445870802188522027398100
