| L(s) = 1 | − 1.18·2-s − 3·3-s − 6.59·4-s − 20.2·5-s + 3.56·6-s − 7·7-s + 17.3·8-s + 9·9-s + 24.0·10-s + 30.2·11-s + 19.7·12-s + 32.2·13-s + 8.31·14-s + 60.6·15-s + 32.1·16-s + 41.2·17-s − 10.6·18-s − 51.4·19-s + 133.·20-s + 21·21-s − 35.9·22-s + 23·23-s − 51.9·24-s + 284.·25-s − 38.2·26-s − 27·27-s + 46.1·28-s + ⋯ |
| L(s) = 1 | − 0.419·2-s − 0.577·3-s − 0.823·4-s − 1.80·5-s + 0.242·6-s − 0.377·7-s + 0.765·8-s + 0.333·9-s + 0.759·10-s + 0.830·11-s + 0.475·12-s + 0.687·13-s + 0.158·14-s + 1.04·15-s + 0.502·16-s + 0.589·17-s − 0.139·18-s − 0.620·19-s + 1.49·20-s + 0.218·21-s − 0.348·22-s + 0.208·23-s − 0.441·24-s + 2.27·25-s − 0.288·26-s − 0.192·27-s + 0.311·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 + 1.18T + 8T^{2} \) |
| 5 | \( 1 + 20.2T + 125T^{2} \) |
| 11 | \( 1 - 30.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 32.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 41.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 51.4T + 6.85e3T^{2} \) |
| 29 | \( 1 + 142.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 174.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 417.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 183.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 465.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 578.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 263.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 161.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 204.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 420.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 107.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 518.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 390.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 102.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 6.21T + 7.04e5T^{2} \) |
| 97 | \( 1 - 733.T + 9.12e5T^{2} \) |
| 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
−10.22563183293232621487686019984, −9.037112989359849562218765187380, −8.397396104050940865782138180980, −7.50179813296686946567922927008, −6.60006368585232467358096481937, −5.20926151218106909688988624341, −4.08783487372900512233113431748, −3.60053822835744903506064209959, −1.04056236721153631505817717464, 0,
1.04056236721153631505817717464, 3.60053822835744903506064209959, 4.08783487372900512233113431748, 5.20926151218106909688988624341, 6.60006368585232467358096481937, 7.50179813296686946567922927008, 8.397396104050940865782138180980, 9.037112989359849562218765187380, 10.22563183293232621487686019984
