| L(s) = 1 | + 3.59·2-s − 4.68·3-s + 4.94·4-s − 3.25·5-s − 16.8·6-s + 7·7-s − 10.9·8-s − 5.03·9-s − 11.7·10-s − 52.7·11-s − 23.1·12-s − 2.09·13-s + 25.1·14-s + 15.2·15-s − 79.1·16-s − 14.6·17-s − 18.1·18-s + 6.38·19-s − 16.1·20-s − 32.8·21-s − 189.·22-s − 23·23-s + 51.5·24-s − 114.·25-s − 7.54·26-s + 150.·27-s + 34.6·28-s + ⋯ |
| L(s) = 1 | + 1.27·2-s − 0.901·3-s + 0.617·4-s − 0.291·5-s − 1.14·6-s + 0.377·7-s − 0.485·8-s − 0.186·9-s − 0.370·10-s − 1.44·11-s − 0.557·12-s − 0.0447·13-s + 0.480·14-s + 0.262·15-s − 1.23·16-s − 0.208·17-s − 0.237·18-s + 0.0770·19-s − 0.180·20-s − 0.340·21-s − 1.83·22-s − 0.208·23-s + 0.438·24-s − 0.915·25-s − 0.0569·26-s + 1.07·27-s + 0.233·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 - 7T \) |
| 23 | \( 1 + 23T \) |
| good | 2 | \( 1 - 3.59T + 8T^{2} \) |
| 3 | \( 1 + 4.68T + 27T^{2} \) |
| 5 | \( 1 + 3.25T + 125T^{2} \) |
| 11 | \( 1 + 52.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 2.09T + 2.19e3T^{2} \) |
| 17 | \( 1 + 14.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 6.38T + 6.85e3T^{2} \) |
| 29 | \( 1 - 101.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 96.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 219.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 462.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 148.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 16.0T + 1.03e5T^{2} \) |
| 53 | \( 1 - 408.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 194.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 272.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 811.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 867.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 948.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 149.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 824.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 370.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.64e3T + 9.12e5T^{2} \) |
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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
−12.02406528810075461391662929152, −11.27517870330097411214719799473, −10.33628988617405612438342974779, −8.700143603857458454532589729990, −7.40490681274235955905233320692, −5.95502731641321939422691887783, −5.30817924700667502704616589378, −4.27284430159394407740732167527, −2.71018609025086626802747110317, 0,
2.71018609025086626802747110317, 4.27284430159394407740732167527, 5.30817924700667502704616589378, 5.95502731641321939422691887783, 7.40490681274235955905233320692, 8.700143603857458454532589729990, 10.33628988617405612438342974779, 11.27517870330097411214719799473, 12.02406528810075461391662929152
