| L(s) = 1 | − 3.93·2-s + 7.47·4-s − 2.43·5-s − 7·7-s + 2.05·8-s + 9.56·10-s + 42.1·11-s − 39.5·13-s + 27.5·14-s − 67.9·16-s − 2.07·17-s + 96.1·19-s − 18.1·20-s − 165.·22-s + 73.7·23-s − 119.·25-s + 155.·26-s − 52.3·28-s + 19.2·29-s − 239.·31-s + 250.·32-s + 8.17·34-s + 17.0·35-s − 144.·37-s − 378.·38-s − 4.99·40-s − 72.2·41-s + ⋯ |
| L(s) = 1 | − 1.39·2-s + 0.934·4-s − 0.217·5-s − 0.377·7-s + 0.0908·8-s + 0.302·10-s + 1.15·11-s − 0.844·13-s + 0.525·14-s − 1.06·16-s − 0.0296·17-s + 1.16·19-s − 0.203·20-s − 1.60·22-s + 0.668·23-s − 0.952·25-s + 1.17·26-s − 0.353·28-s + 0.122·29-s − 1.38·31-s + 1.38·32-s + 0.0412·34-s + 0.0821·35-s − 0.641·37-s − 1.61·38-s − 0.0197·40-s − 0.275·41-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
| \(L(2)\) |
\(\approx\) |
\(0.7434069530\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7434069530\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 3.93T + 8T^{2} \) |
| 5 | \( 1 + 2.43T + 125T^{2} \) |
| 11 | \( 1 - 42.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 39.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 2.07T + 4.91e3T^{2} \) |
| 19 | \( 1 - 96.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 73.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 19.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 239.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 144.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 72.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 480.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 294.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 627.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 149.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 630.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 4.04T + 3.00e5T^{2} \) |
| 71 | \( 1 + 798.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 444.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 575.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.29e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 750.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 419.T + 9.12e5T^{2} \) |
| 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
−10.07035034751039017890092614587, −9.307439167666216838127226184551, −8.920565491967251745653753536737, −7.56933783672354981437120433052, −7.24517161801854912114630075488, −6.03591880313469863144149224246, −4.67084972246582077022113389998, −3.40856860428456535100845276515, −1.88751806096404699363686432157, −0.64839473736101196123797063301,
0.64839473736101196123797063301, 1.88751806096404699363686432157, 3.40856860428456535100845276515, 4.67084972246582077022113389998, 6.03591880313469863144149224246, 7.24517161801854912114630075488, 7.56933783672354981437120433052, 8.920565491967251745653753536737, 9.307439167666216838127226184551, 10.07035034751039017890092614587
