| L(s) = 1 | + 4.58·2-s − 3·3-s + 12.9·4-s − 4.89·5-s − 13.7·6-s + 1.55·7-s + 22.9·8-s + 9·9-s − 22.4·10-s − 31.1·11-s − 38.9·12-s − 4.75·13-s + 7.12·14-s + 14.6·15-s + 0.976·16-s − 11.4·17-s + 41.2·18-s − 41.8·19-s − 63.6·20-s − 4.66·21-s − 142.·22-s + 123.·23-s − 68.7·24-s − 101.·25-s − 21.7·26-s − 27·27-s + 20.2·28-s + ⋯ |
| L(s) = 1 | + 1.62·2-s − 0.577·3-s + 1.62·4-s − 0.438·5-s − 0.935·6-s + 0.0839·7-s + 1.01·8-s + 0.333·9-s − 0.709·10-s − 0.853·11-s − 0.938·12-s − 0.101·13-s + 0.136·14-s + 0.252·15-s + 0.0152·16-s − 0.162·17-s + 0.540·18-s − 0.505·19-s − 0.711·20-s − 0.0484·21-s − 1.38·22-s + 1.11·23-s − 0.584·24-s − 0.808·25-s − 0.164·26-s − 0.192·27-s + 0.136·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 211 | \( 1 - 211T \) |
| good | 2 | \( 1 - 4.58T + 8T^{2} \) |
| 5 | \( 1 + 4.89T + 125T^{2} \) |
| 7 | \( 1 - 1.55T + 343T^{2} \) |
| 11 | \( 1 + 31.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 4.75T + 2.19e3T^{2} \) |
| 17 | \( 1 + 11.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 41.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 123.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 257.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 71.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 98.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 340.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 5.98T + 7.95e4T^{2} \) |
| 47 | \( 1 - 284.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 339.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 211.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 522.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 621.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 832.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 280.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 276.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 145.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 882.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.66e3T + 9.12e5T^{2} \) |
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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
−10.11837744724114244714133837717, −8.813201894456933834893755194468, −7.55440999919825396872158788551, −6.84274545011374334434344041004, −5.74272734842348649003218281882, −5.15058697992011834095035934613, −4.22107300646247156841493840720, −3.30017368100209084338610262801, −2.02997828545131423988634758383, 0,
2.02997828545131423988634758383, 3.30017368100209084338610262801, 4.22107300646247156841493840720, 5.15058697992011834095035934613, 5.74272734842348649003218281882, 6.84274545011374334434344041004, 7.55440999919825396872158788551, 8.813201894456933834893755194468, 10.11837744724114244714133837717
