| L(s) = 1 | − 10.9·2-s + 9·3-s + 88.6·4-s − 108.·5-s − 98.8·6-s − 49·7-s − 622.·8-s + 81·9-s + 1.19e3·10-s − 469.·11-s + 797.·12-s − 1.00e3·13-s + 538.·14-s − 980.·15-s + 3.99e3·16-s + 529.·17-s − 889.·18-s + 2.06e3·19-s − 9.65e3·20-s − 441·21-s + 5.15e3·22-s + 529·23-s − 5.59e3·24-s + 8.74e3·25-s + 1.10e4·26-s + 729·27-s − 4.34e3·28-s + ⋯ |
| L(s) = 1 | − 1.94·2-s + 0.577·3-s + 2.76·4-s − 1.94·5-s − 1.12·6-s − 0.377·7-s − 3.43·8-s + 0.333·9-s + 3.78·10-s − 1.16·11-s + 1.59·12-s − 1.65·13-s + 0.733·14-s − 1.12·15-s + 3.90·16-s + 0.444·17-s − 0.647·18-s + 1.31·19-s − 5.39·20-s − 0.218·21-s + 2.27·22-s + 0.208·23-s − 1.98·24-s + 2.79·25-s + 3.20·26-s + 0.192·27-s − 1.04·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 7 | \( 1 + 49T \) |
| 23 | \( 1 - 529T \) |
| good | 2 | \( 1 + 10.9T + 32T^{2} \) |
| 5 | \( 1 + 108.T + 3.12e3T^{2} \) |
| 11 | \( 1 + 469.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.00e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 529.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.06e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 7.40e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.93e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.76e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.41e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 7.99e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.24e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.80e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.05e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.53e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 8.93e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.89e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.29e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.58e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.03e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.30e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.65e4T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.802810542370581298698440347373, −8.593665208833871110121586916106, −8.008116643704571391499477477812, −7.44730001358382552002114216847, −6.86899258840350065834998875360, −4.94918861900733780290591429545, −3.20806372899656070395424029609, −2.66590208833874356034089013621, −0.859475189417492961563328612813, 0,
0.859475189417492961563328612813, 2.66590208833874356034089013621, 3.20806372899656070395424029609, 4.94918861900733780290591429545, 6.86899258840350065834998875360, 7.44730001358382552002114216847, 8.008116643704571391499477477812, 8.593665208833871110121586916106, 9.802810542370581298698440347373
