| L(s) = 1 | + 3.75·2-s − 3·3-s + 6.13·4-s + 3.12·5-s − 11.2·6-s − 7·7-s − 7.01·8-s + 9·9-s + 11.7·10-s + 5.61·11-s − 18.4·12-s + 20.5·13-s − 26.3·14-s − 9.36·15-s − 75.4·16-s − 86.5·17-s + 33.8·18-s − 81.4·19-s + 19.1·20-s + 21·21-s + 21.1·22-s + 23·23-s + 21.0·24-s − 115.·25-s + 77.4·26-s − 27·27-s − 42.9·28-s + ⋯ |
| L(s) = 1 | + 1.32·2-s − 0.577·3-s + 0.766·4-s + 0.279·5-s − 0.767·6-s − 0.377·7-s − 0.309·8-s + 0.333·9-s + 0.371·10-s + 0.153·11-s − 0.442·12-s + 0.439·13-s − 0.502·14-s − 0.161·15-s − 1.17·16-s − 1.23·17-s + 0.443·18-s − 0.982·19-s + 0.214·20-s + 0.218·21-s + 0.204·22-s + 0.208·23-s + 0.178·24-s − 0.922·25-s + 0.583·26-s − 0.192·27-s − 0.289·28-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 | \( 1 - 3.75T + 8T^{2} \) |
| 5 | \( 1 - 3.12T + 125T^{2} \) |
| 11 | \( 1 - 5.61T + 1.33e3T^{2} \) |
| 13 | \( 1 - 20.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 86.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 81.4T + 6.85e3T^{2} \) |
| 29 | \( 1 + 39.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 244.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 309.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 50.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 423.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 599.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 160.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 194.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 580.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 737.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 771.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 377.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.04e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 389.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.11e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 513.T + 9.12e5T^{2} \) |
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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
−10.36170394870844975969324287021, −9.319649814445596465993622583094, −8.314688276927199884927637519409, −6.60146119863009519182241374509, −6.41425691677463291134621870728, −5.24596591958989951443172550416, −4.41355864943864829952012893576, −3.42566940545908129455225094714, −2.03566091531373066595773156555, 0,
2.03566091531373066595773156555, 3.42566940545908129455225094714, 4.41355864943864829952012893576, 5.24596591958989951443172550416, 6.41425691677463291134621870728, 6.60146119863009519182241374509, 8.314688276927199884927637519409, 9.319649814445596465993622583094, 10.36170394870844975969324287021
