| L(s) = 1 | − 2-s − 4-s + 5-s − 1.73·7-s + 3·8-s − 10-s + 2.46·11-s − 13-s + 1.73·14-s − 16-s − 4·17-s − 3.73·19-s − 20-s − 2.46·22-s + 2.53·23-s + 25-s + 26-s + 1.73·28-s + 9.19·29-s − 3.46·31-s − 5·32-s + 4·34-s − 1.73·35-s + 5.46·37-s + 3.73·38-s + 3·40-s − 5.46·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.5·4-s + 0.447·5-s − 0.654·7-s + 1.06·8-s − 0.316·10-s + 0.742·11-s − 0.277·13-s + 0.462·14-s − 0.250·16-s − 0.970·17-s − 0.856·19-s − 0.223·20-s − 0.525·22-s + 0.528·23-s + 0.200·25-s + 0.196·26-s + 0.327·28-s + 1.70·29-s − 0.622·31-s − 0.883·32-s + 0.685·34-s − 0.292·35-s + 0.898·37-s + 0.605·38-s + 0.474·40-s − 0.853·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + T + 2T^{2} \) |
| 7 | \( 1 + 1.73T + 7T^{2} \) |
| 11 | \( 1 - 2.46T + 11T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 3.73T + 19T^{2} \) |
| 23 | \( 1 - 2.53T + 23T^{2} \) |
| 29 | \( 1 - 9.19T + 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 - 5.46T + 37T^{2} \) |
| 41 | \( 1 + 5.46T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 - 2.92T + 47T^{2} \) |
| 53 | \( 1 - 7.19T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 0.0717T + 61T^{2} \) |
| 67 | \( 1 + 3.46T + 67T^{2} \) |
| 71 | \( 1 + 7.53T + 71T^{2} \) |
| 73 | \( 1 - 1.46T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 3.92T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−8.093693811478785622721346457464, −7.01019856300183623012608516629, −6.62100399725221277459996420420, −5.78895111168216162029304337183, −4.73665583604702597303963660869, −4.27931488586637484798036076356, −3.22532281709913286312703719710, −2.19877234324633748590927765277, −1.17084287467558862676802162761, 0,
1.17084287467558862676802162761, 2.19877234324633748590927765277, 3.22532281709913286312703719710, 4.27931488586637484798036076356, 4.73665583604702597303963660869, 5.78895111168216162029304337183, 6.62100399725221277459996420420, 7.01019856300183623012608516629, 8.093693811478785622721346457464
