| L(s) = 1 | + 7.91·2-s + 12.8·3-s + 30.6·4-s + 79.5·5-s + 101.·6-s − 172.·7-s − 11.0·8-s − 77.1·9-s + 629.·10-s + 452.·11-s + 394.·12-s − 22.7·13-s − 1.36e3·14-s + 1.02e3·15-s − 1.06e3·16-s − 521.·17-s − 610.·18-s + 1.55e3·19-s + 2.43e3·20-s − 2.21e3·21-s + 3.57e3·22-s − 3.46e3·23-s − 142.·24-s + 3.20e3·25-s − 179.·26-s − 4.12e3·27-s − 5.27e3·28-s + ⋯ |
| L(s) = 1 | + 1.39·2-s + 0.826·3-s + 0.956·4-s + 1.42·5-s + 1.15·6-s − 1.32·7-s − 0.0610·8-s − 0.317·9-s + 1.99·10-s + 1.12·11-s + 0.790·12-s − 0.0373·13-s − 1.85·14-s + 1.17·15-s − 1.04·16-s − 0.437·17-s − 0.443·18-s + 0.990·19-s + 1.36·20-s − 1.09·21-s + 1.57·22-s − 1.36·23-s − 0.0504·24-s + 1.02·25-s − 0.0522·26-s − 1.08·27-s − 1.27·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.140209250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.140209250\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 - 1.84e3T \) |
| good | 2 | \( 1 - 7.91T + 32T^{2} \) |
| 3 | \( 1 - 12.8T + 243T^{2} \) |
| 5 | \( 1 - 79.5T + 3.12e3T^{2} \) |
| 7 | \( 1 + 172.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 452.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 22.7T + 3.71e5T^{2} \) |
| 17 | \( 1 + 521.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.55e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.46e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.32e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.98e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.00e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.64e4T + 1.15e8T^{2} \) |
| 47 | \( 1 - 2.41e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.12e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.58e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.85e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.67e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.00e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.21e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.19e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.67e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.89e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.30e4T + 8.58e9T^{2} \) |
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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
−14.34658625523888390900322087240, −13.86641445316528797197803290656, −13.09172048736677320018229075088, −11.88586960105509747869882795605, −9.824430344986317449086235401161, −9.036092422463436472160240890261, −6.58325898323565812450751081411, −5.70944105790103814484252456062, −3.69938247848136015406298073891, −2.44971709892072270243620974604,
2.44971709892072270243620974604, 3.69938247848136015406298073891, 5.70944105790103814484252456062, 6.58325898323565812450751081411, 9.036092422463436472160240890261, 9.824430344986317449086235401161, 11.88586960105509747869882795605, 13.09172048736677320018229075088, 13.86641445316528797197803290656, 14.34658625523888390900322087240
