| L(s) = 1 | − 19.5·2-s − 3.33·3-s + 252.·4-s + 170.·5-s + 65.0·6-s + 31.4·7-s − 2.43e3·8-s − 2.17e3·9-s − 3.32e3·10-s − 3.82e3·11-s − 842.·12-s + 3.30e3·13-s − 614.·14-s − 568.·15-s + 1.51e4·16-s − 2.46e4·17-s + 4.24e4·18-s + 6.85e3·19-s + 4.30e4·20-s − 104.·21-s + 7.46e4·22-s − 9.14e4·23-s + 8.12e3·24-s − 4.90e4·25-s − 6.44e4·26-s + 1.45e4·27-s + 7.95e3·28-s + ⋯ |
| L(s) = 1 | − 1.72·2-s − 0.0712·3-s + 1.97·4-s + 0.609·5-s + 0.122·6-s + 0.0346·7-s − 1.68·8-s − 0.994·9-s − 1.05·10-s − 0.866·11-s − 0.140·12-s + 0.416·13-s − 0.0598·14-s − 0.0434·15-s + 0.926·16-s − 1.21·17-s + 1.71·18-s + 0.229·19-s + 1.20·20-s − 0.00247·21-s + 1.49·22-s − 1.56·23-s + 0.119·24-s − 0.628·25-s − 0.719·26-s + 0.142·27-s + 0.0685·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 - 6.85e3T \) |
| good | 2 | \( 1 + 19.5T + 128T^{2} \) |
| 3 | \( 1 + 3.33T + 2.18e3T^{2} \) |
| 5 | \( 1 - 170.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 31.4T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.82e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 3.30e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.46e4T + 4.10e8T^{2} \) |
| 23 | \( 1 + 9.14e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.20e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 4.51e3T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.18e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.48e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.45e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.21e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.16e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.75e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 4.39e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 6.26e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 6.20e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.13e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.23e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.68e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.50e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 5.34e6T + 8.07e13T^{2} \) |
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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
−16.67524671240004364107127235131, −15.49108406178130475658287700068, −13.61563746263433424660339963730, −11.52995355556944873493640779296, −10.39015559826478387606945877457, −9.101066792903639275410024048666, −7.891752060594996440077031514104, −6.05952201512752652457829302237, −2.19378523249737211995813802187, 0,
2.19378523249737211995813802187, 6.05952201512752652457829302237, 7.891752060594996440077031514104, 9.101066792903639275410024048666, 10.39015559826478387606945877457, 11.52995355556944873493640779296, 13.61563746263433424660339963730, 15.49108406178130475658287700068, 16.67524671240004364107127235131
