L(s) = 1 | + 0.807·2-s − 7.34·4-s + 18.2·5-s − 7·7-s − 12.3·8-s + 14.7·10-s − 49.4·11-s + 44.4·13-s − 5.65·14-s + 48.7·16-s + 47.2·17-s + 56.7·19-s − 133.·20-s − 39.9·22-s − 55.4·23-s + 207.·25-s + 35.9·26-s + 51.4·28-s + 150.·29-s − 167.·31-s + 138.·32-s + 38.1·34-s − 127.·35-s + 331.·37-s + 45.8·38-s − 225.·40-s + 295.·41-s + ⋯ |
L(s) = 1 | + 0.285·2-s − 0.918·4-s + 1.63·5-s − 0.377·7-s − 0.547·8-s + 0.465·10-s − 1.35·11-s + 0.949·13-s − 0.107·14-s + 0.762·16-s + 0.673·17-s + 0.685·19-s − 1.49·20-s − 0.386·22-s − 0.502·23-s + 1.65·25-s + 0.270·26-s + 0.347·28-s + 0.962·29-s − 0.970·31-s + 0.765·32-s + 0.192·34-s − 0.616·35-s + 1.47·37-s + 0.195·38-s − 0.892·40-s + 1.12·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.291951215\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.291951215\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
good | 2 | \( 1 - 0.807T + 8T^{2} \) |
| 5 | \( 1 - 18.2T + 125T^{2} \) |
| 11 | \( 1 + 49.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 44.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 47.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 56.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 55.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 150.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 167.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 331.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 295.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 317.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 137.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 411.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 212.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 52.3T + 2.26e5T^{2} \) |
| 67 | \( 1 - 706.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 78.7T + 3.57e5T^{2} \) |
| 73 | \( 1 - 839.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.01e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.08e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 762.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.34e3T + 9.12e5T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.01112299547976561420205120756, −9.702268294250300235283971760278, −8.718437056256337295642065422119, −7.80739951212169852179757687182, −6.32564299820104854278790889700, −5.63565389972906301315918629928, −5.00118678792501349202734527756, −3.54521156394464639328599815003, −2.43796025571670008641189145756, −0.906559265284840969815175048477,
0.906559265284840969815175048477, 2.43796025571670008641189145756, 3.54521156394464639328599815003, 5.00118678792501349202734527756, 5.63565389972906301315918629928, 6.32564299820104854278790889700, 7.80739951212169852179757687182, 8.718437056256337295642065422119, 9.702268294250300235283971760278, 10.01112299547976561420205120756