| L(s) = 1 | − 4.74·2-s − 3·3-s + 14.4·4-s − 4.51·5-s + 14.2·6-s + 7.48·7-s − 30.7·8-s + 9·9-s + 21.4·10-s + 66.8·11-s − 43.4·12-s − 35.4·14-s + 13.5·15-s + 29.8·16-s + 96.9·17-s − 42.6·18-s − 31.4·19-s − 65.4·20-s − 22.4·21-s − 317.·22-s + 183.·23-s + 92.2·24-s − 104.·25-s − 27·27-s + 108.·28-s + 112.·29-s − 64.2·30-s + ⋯ |
| L(s) = 1 | − 1.67·2-s − 0.577·3-s + 1.81·4-s − 0.403·5-s + 0.967·6-s + 0.404·7-s − 1.35·8-s + 0.333·9-s + 0.677·10-s + 1.83·11-s − 1.04·12-s − 0.677·14-s + 0.233·15-s + 0.467·16-s + 1.38·17-s − 0.558·18-s − 0.380·19-s − 0.731·20-s − 0.233·21-s − 3.07·22-s + 1.66·23-s + 0.784·24-s − 0.836·25-s − 0.192·27-s + 0.731·28-s + 0.718·29-s − 0.391·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7874530294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7874530294\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 4.74T + 8T^{2} \) |
| 5 | \( 1 + 4.51T + 125T^{2} \) |
| 7 | \( 1 - 7.48T + 343T^{2} \) |
| 11 | \( 1 - 66.8T + 1.33e3T^{2} \) |
| 17 | \( 1 - 96.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 31.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 183.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 112.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 77.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 54.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 451.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 113.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 42.2T + 1.03e5T^{2} \) |
| 53 | \( 1 + 530.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 219.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 822.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 872.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 100.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 165.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 545.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 454.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 230.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.08e3T + 9.12e5T^{2} \) |
| 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
−10.29595870313518155911948904042, −9.625011469330086613022948104963, −8.743265685928129015153887863074, −7.985683677464502023949907952137, −6.99315096921287786618545786195, −6.36607898130008163948285387391, −4.89393975198094449727723990123, −3.48941743535896851954136006874, −1.62345701760583138515702463091, −0.792805486461538981239217432552,
0.792805486461538981239217432552, 1.62345701760583138515702463091, 3.48941743535896851954136006874, 4.89393975198094449727723990123, 6.36607898130008163948285387391, 6.99315096921287786618545786195, 7.985683677464502023949907952137, 8.743265685928129015153887863074, 9.625011469330086613022948104963, 10.29595870313518155911948904042
