| L(s) = 1 | − 5.43·2-s + 3·3-s + 21.5·4-s − 5.05·5-s − 16.3·6-s − 7·7-s − 73.7·8-s + 9·9-s + 27.5·10-s + 7.71·11-s + 64.6·12-s + 24.7·13-s + 38.0·14-s − 15.1·15-s + 228.·16-s − 50.1·17-s − 48.9·18-s + 69.7·19-s − 109.·20-s − 21·21-s − 41.9·22-s − 23·23-s − 221.·24-s − 99.4·25-s − 134.·26-s + 27·27-s − 150.·28-s + ⋯ |
| L(s) = 1 | − 1.92·2-s + 0.577·3-s + 2.69·4-s − 0.452·5-s − 1.10·6-s − 0.377·7-s − 3.25·8-s + 0.333·9-s + 0.869·10-s + 0.211·11-s + 1.55·12-s + 0.527·13-s + 0.726·14-s − 0.261·15-s + 3.57·16-s − 0.715·17-s − 0.640·18-s + 0.842·19-s − 1.21·20-s − 0.218·21-s − 0.406·22-s − 0.208·23-s − 1.88·24-s − 0.795·25-s − 1.01·26-s + 0.192·27-s − 1.01·28-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 | \( 1 + 5.43T + 8T^{2} \) |
| 5 | \( 1 + 5.05T + 125T^{2} \) |
| 11 | \( 1 - 7.71T + 1.33e3T^{2} \) |
| 13 | \( 1 - 24.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 50.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 69.7T + 6.85e3T^{2} \) |
| 29 | \( 1 + 108.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 99.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 179.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 213.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 113.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 360.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 41.2T + 1.48e5T^{2} \) |
| 59 | \( 1 + 105.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 135.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 709.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 415.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 25.2T + 3.89e5T^{2} \) |
| 79 | \( 1 - 594.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 989.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 794.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.24e3T + 9.12e5T^{2} \) |
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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
−9.719214144520270024571239477205, −9.307776641810691886305053088497, −8.386744070931594614532974478605, −7.69096523871990688894608887799, −6.91877035016684721133303889249, −5.90578932915289355626436217693, −3.80931494080589566886561945261, −2.63570723774586788310220868508, −1.39997564398031630749336138544, 0,
1.39997564398031630749336138544, 2.63570723774586788310220868508, 3.80931494080589566886561945261, 5.90578932915289355626436217693, 6.91877035016684721133303889249, 7.69096523871990688894608887799, 8.386744070931594614532974478605, 9.307776641810691886305053088497, 9.719214144520270024571239477205
