| L(s) = 1 | − 2-s + 0.854·3-s + 4-s + 5-s − 0.854·6-s + 0.252·7-s − 8-s − 2.26·9-s − 10-s + 2.77·11-s + 0.854·12-s − 13-s − 0.252·14-s + 0.854·15-s + 16-s − 6.26·17-s + 2.26·18-s + 5.36·19-s + 20-s + 0.215·21-s − 2.77·22-s − 9.02·23-s − 0.854·24-s + 25-s + 26-s − 4.50·27-s + 0.252·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.493·3-s + 0.5·4-s + 0.447·5-s − 0.348·6-s + 0.0954·7-s − 0.353·8-s − 0.756·9-s − 0.316·10-s + 0.836·11-s + 0.246·12-s − 0.277·13-s − 0.0675·14-s + 0.220·15-s + 0.250·16-s − 1.51·17-s + 0.534·18-s + 1.23·19-s + 0.223·20-s + 0.0471·21-s − 0.591·22-s − 1.88·23-s − 0.174·24-s + 0.200·25-s + 0.196·26-s − 0.866·27-s + 0.0477·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 3 | \( 1 - 0.854T + 3T^{2} \) |
| 7 | \( 1 - 0.252T + 7T^{2} \) |
| 11 | \( 1 - 2.77T + 11T^{2} \) |
| 17 | \( 1 + 6.26T + 17T^{2} \) |
| 19 | \( 1 - 5.36T + 19T^{2} \) |
| 23 | \( 1 + 9.02T + 23T^{2} \) |
| 29 | \( 1 - 0.739T + 29T^{2} \) |
| 37 | \( 1 - 1.20T + 37T^{2} \) |
| 41 | \( 1 - 3.51T + 41T^{2} \) |
| 43 | \( 1 - 1.65T + 43T^{2} \) |
| 47 | \( 1 + 9.56T + 47T^{2} \) |
| 53 | \( 1 + 1.77T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 + 2.52T + 61T^{2} \) |
| 67 | \( 1 + 5.00T + 67T^{2} \) |
| 71 | \( 1 - 3.84T + 71T^{2} \) |
| 73 | \( 1 + 0.875T + 73T^{2} \) |
| 79 | \( 1 - 0.848T + 79T^{2} \) |
| 83 | \( 1 - 5.40T + 83T^{2} \) |
| 89 | \( 1 - 7.15T + 89T^{2} \) |
| 97 | \( 1 - 4.06T + 97T^{2} \) |
| 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
−8.074131524881025751255455235764, −7.64600931534547871336020345083, −6.50044460631065509087818948884, −6.18821259857422880627738615933, −5.15066083882354424574183820191, −4.15619555042068840657227010181, −3.17777419529141123703995732019, −2.32768584874165890983556832202, −1.52737485071282497940604692310, 0,
1.52737485071282497940604692310, 2.32768584874165890983556832202, 3.17777419529141123703995732019, 4.15619555042068840657227010181, 5.15066083882354424574183820191, 6.18821259857422880627738615933, 6.50044460631065509087818948884, 7.64600931534547871336020345083, 8.074131524881025751255455235764
