| L(s) = 1 | − 2-s − 0.672·3-s + 4-s − 2.38·5-s + 0.672·6-s − 0.506·7-s − 8-s − 2.54·9-s + 2.38·10-s + 1.82·11-s − 0.672·12-s + 1.61·13-s + 0.506·14-s + 1.60·15-s + 16-s − 5.88·17-s + 2.54·18-s − 3.78·19-s − 2.38·20-s + 0.340·21-s − 1.82·22-s − 23-s + 0.672·24-s + 0.681·25-s − 1.61·26-s + 3.73·27-s − 0.506·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.388·3-s + 0.5·4-s − 1.06·5-s + 0.274·6-s − 0.191·7-s − 0.353·8-s − 0.849·9-s + 0.753·10-s + 0.549·11-s − 0.194·12-s + 0.446·13-s + 0.135·14-s + 0.414·15-s + 0.250·16-s − 1.42·17-s + 0.600·18-s − 0.868·19-s − 0.532·20-s + 0.0743·21-s − 0.388·22-s − 0.208·23-s + 0.137·24-s + 0.136·25-s − 0.316·26-s + 0.718·27-s − 0.0957·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 + 0.672T + 3T^{2} \) |
| 5 | \( 1 + 2.38T + 5T^{2} \) |
| 7 | \( 1 + 0.506T + 7T^{2} \) |
| 11 | \( 1 - 1.82T + 11T^{2} \) |
| 13 | \( 1 - 1.61T + 13T^{2} \) |
| 17 | \( 1 + 5.88T + 17T^{2} \) |
| 19 | \( 1 + 3.78T + 19T^{2} \) |
| 29 | \( 1 - 9.57T + 29T^{2} \) |
| 31 | \( 1 - 4.60T + 31T^{2} \) |
| 37 | \( 1 - 8.91T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 + 6.39T + 43T^{2} \) |
| 47 | \( 1 + 1.18T + 47T^{2} \) |
| 53 | \( 1 - 2.70T + 53T^{2} \) |
| 59 | \( 1 - 8.49T + 59T^{2} \) |
| 61 | \( 1 + 9.41T + 61T^{2} \) |
| 67 | \( 1 - 9.70T + 67T^{2} \) |
| 71 | \( 1 + 7.20T + 71T^{2} \) |
| 73 | \( 1 - 0.843T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 - 3.54T + 83T^{2} \) |
| 89 | \( 1 + 8.85T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
| 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
−8.018074787046628885112357723308, −6.93235533708935784698738228345, −6.42656628864834642585585908518, −5.90638591185530534239294171738, −4.60087067389798947884020150671, −4.17643161875427757657309103448, −3.09352172978608081022867790454, −2.33387962916475461815415695420, −0.938800171414450475986711813746, 0,
0.938800171414450475986711813746, 2.33387962916475461815415695420, 3.09352172978608081022867790454, 4.17643161875427757657309103448, 4.60087067389798947884020150671, 5.90638591185530534239294171738, 6.42656628864834642585585908518, 6.93235533708935784698738228345, 8.018074787046628885112357723308
