| L(s) = 1 | − 3.69·2-s + 5.63·4-s + 18.7·5-s + 24.1·7-s + 8.74·8-s − 69.2·10-s + 49.2·11-s − 89.1·14-s − 77.3·16-s − 65.3·17-s − 109.·19-s + 105.·20-s − 181.·22-s − 83.2·23-s + 226.·25-s + 135.·28-s + 4.99·29-s + 255.·31-s + 215.·32-s + 241.·34-s + 452.·35-s − 93.6·37-s + 405.·38-s + 164.·40-s − 67.9·41-s + 142.·43-s + 277.·44-s + ⋯ |
| L(s) = 1 | − 1.30·2-s + 0.703·4-s + 1.67·5-s + 1.30·7-s + 0.386·8-s − 2.19·10-s + 1.35·11-s − 1.70·14-s − 1.20·16-s − 0.931·17-s − 1.32·19-s + 1.18·20-s − 1.76·22-s − 0.755·23-s + 1.81·25-s + 0.917·28-s + 0.0319·29-s + 1.48·31-s + 1.19·32-s + 1.21·34-s + 2.18·35-s − 0.416·37-s + 1.73·38-s + 0.648·40-s − 0.258·41-s + 0.505·43-s + 0.950·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.958359107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.958359107\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 3.69T + 8T^{2} \) |
| 5 | \( 1 - 18.7T + 125T^{2} \) |
| 7 | \( 1 - 24.1T + 343T^{2} \) |
| 11 | \( 1 - 49.2T + 1.33e3T^{2} \) |
| 17 | \( 1 + 65.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 109.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 83.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 4.99T + 2.43e4T^{2} \) |
| 31 | \( 1 - 255.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 93.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 67.9T + 6.89e4T^{2} \) |
| 43 | \( 1 - 142.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 379.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 389.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 133.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 620.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 119.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 361.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 748.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 514.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 260.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 833.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.48e3T + 9.12e5T^{2} \) |
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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
−8.891590969959219424190024662308, −8.733097265766528631528638340305, −7.75491422880212030498606795774, −6.62143471518436709445593817068, −6.19903147624109285976622005187, −4.93157166102720854810517446926, −4.22558254250489986520412408538, −2.21537667302448073333053445472, −1.81901921709057545725221894041, −0.887138585401610032837980828928,
0.887138585401610032837980828928, 1.81901921709057545725221894041, 2.21537667302448073333053445472, 4.22558254250489986520412408538, 4.93157166102720854810517446926, 6.19903147624109285976622005187, 6.62143471518436709445593817068, 7.75491422880212030498606795774, 8.733097265766528631528638340305, 8.891590969959219424190024662308
