| L(s) = 1 | − 2-s + 3-s + 4-s + 0.137·5-s − 6-s + 0.345·7-s − 8-s + 9-s − 0.137·10-s + 0.938·11-s + 12-s − 3.67·13-s − 0.345·14-s + 0.137·15-s + 16-s + 17-s − 18-s + 4.36·19-s + 0.137·20-s + 0.345·21-s − 0.938·22-s + 7.24·23-s − 24-s − 4.98·25-s + 3.67·26-s + 27-s + 0.345·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.0614·5-s − 0.408·6-s + 0.130·7-s − 0.353·8-s + 0.333·9-s − 0.0434·10-s + 0.283·11-s + 0.288·12-s − 1.02·13-s − 0.0923·14-s + 0.0354·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s + 1.00·19-s + 0.0307·20-s + 0.0754·21-s − 0.200·22-s + 1.51·23-s − 0.204·24-s − 0.996·25-s + 0.721·26-s + 0.192·27-s + 0.0653·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 5 | \( 1 - 0.137T + 5T^{2} \) |
| 7 | \( 1 - 0.345T + 7T^{2} \) |
| 11 | \( 1 - 0.938T + 11T^{2} \) |
| 13 | \( 1 + 3.67T + 13T^{2} \) |
| 19 | \( 1 - 4.36T + 19T^{2} \) |
| 23 | \( 1 - 7.24T + 23T^{2} \) |
| 29 | \( 1 + 6.57T + 29T^{2} \) |
| 31 | \( 1 + 7.20T + 31T^{2} \) |
| 37 | \( 1 + 11.0T + 37T^{2} \) |
| 41 | \( 1 + 9.50T + 41T^{2} \) |
| 43 | \( 1 + 4.77T + 43T^{2} \) |
| 47 | \( 1 + 7.68T + 47T^{2} \) |
| 53 | \( 1 - 6.35T + 53T^{2} \) |
| 61 | \( 1 - 6.03T + 61T^{2} \) |
| 67 | \( 1 - 15.2T + 67T^{2} \) |
| 71 | \( 1 + 9.14T + 71T^{2} \) |
| 73 | \( 1 - 0.323T + 73T^{2} \) |
| 79 | \( 1 - 17.2T + 79T^{2} \) |
| 83 | \( 1 - 2.00T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 - 7.02T + 97T^{2} \) |
| 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
−7.77463293034155002039223629098, −7.10287289049756126586320385937, −6.73378283598066812259602775230, −5.36315544236290099038323635726, −5.11223883396270457457388558756, −3.70648502226376998295595572670, −3.22625949409368919045317052594, −2.12299350922092022455933037248, −1.44968415272618618777489012211, 0,
1.44968415272618618777489012211, 2.12299350922092022455933037248, 3.22625949409368919045317052594, 3.70648502226376998295595572670, 5.11223883396270457457388558756, 5.36315544236290099038323635726, 6.73378283598066812259602775230, 7.10287289049756126586320385937, 7.77463293034155002039223629098
