| L(s) = 1 | + 2-s − 3-s + 4-s + 3.20·5-s − 6-s + 8-s + 9-s + 3.20·10-s + 11-s − 12-s + 6.40·13-s − 3.20·15-s + 16-s − 17-s + 18-s − 1.15·19-s + 3.20·20-s + 22-s + 1.95·23-s − 24-s + 5.24·25-s + 6.40·26-s − 27-s + 7.24·29-s − 3.20·30-s − 10.4·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.43·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s + 1.01·10-s + 0.301·11-s − 0.288·12-s + 1.77·13-s − 0.826·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s − 0.264·19-s + 0.715·20-s + 0.213·22-s + 0.407·23-s − 0.204·24-s + 1.04·25-s + 1.25·26-s − 0.192·27-s + 1.34·29-s − 0.584·30-s − 1.88·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.659032941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.659032941\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 3.20T + 5T^{2} \) |
| 13 | \( 1 - 6.40T + 13T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 19 | \( 1 + 1.15T + 19T^{2} \) |
| 23 | \( 1 - 1.95T + 23T^{2} \) |
| 29 | \( 1 - 7.24T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 - 5.15T + 37T^{2} \) |
| 41 | \( 1 + 8.24T + 41T^{2} \) |
| 43 | \( 1 - 5.15T + 43T^{2} \) |
| 47 | \( 1 - 6.04T + 47T^{2} \) |
| 53 | \( 1 + 6.40T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 + 9.70T + 61T^{2} \) |
| 67 | \( 1 - 6.24T + 67T^{2} \) |
| 71 | \( 1 - 5.24T + 71T^{2} \) |
| 73 | \( 1 - 2.09T + 73T^{2} \) |
| 79 | \( 1 + 5.60T + 79T^{2} \) |
| 83 | \( 1 + 6.55T + 83T^{2} \) |
| 89 | \( 1 - 18.4T + 89T^{2} \) |
| 97 | \( 1 - 5.49T + 97T^{2} \) |
| 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
−8.869720949217232543046609762806, −7.78266293829707790369868885510, −6.67464780534178965950668735202, −6.26982016991356758216391574279, −5.72642615451475320633362944333, −4.97493418352912517494582393066, −4.06029465601077444653575257761, −3.12538746655731341828080346638, −1.96624841020152280249330656048, −1.19143127551374545663880035965,
1.19143127551374545663880035965, 1.96624841020152280249330656048, 3.12538746655731341828080346638, 4.06029465601077444653575257761, 4.97493418352912517494582393066, 5.72642615451475320633362944333, 6.26982016991356758216391574279, 6.67464780534178965950668735202, 7.78266293829707790369868885510, 8.869720949217232543046609762806
