| L(s) = 1 | − 234.·2-s − 187.·3-s + 2.22e4·4-s + 2.26e5·5-s + 4.39e4·6-s − 6.91e5·7-s + 2.47e6·8-s − 1.43e7·9-s − 5.30e7·10-s − 2.43e7·11-s − 4.16e6·12-s + 2.97e7·13-s + 1.62e8·14-s − 4.24e7·15-s − 1.30e9·16-s − 7.77e8·17-s + 3.35e9·18-s + 4.53e9·19-s + 5.02e9·20-s + 1.29e8·21-s + 5.72e9·22-s + 5.86e9·23-s − 4.63e8·24-s + 2.06e10·25-s − 6.98e9·26-s + 5.37e9·27-s − 1.53e10·28-s + ⋯ |
| L(s) = 1 | − 1.29·2-s − 0.0494·3-s + 0.678·4-s + 1.29·5-s + 0.0641·6-s − 0.317·7-s + 0.416·8-s − 0.997·9-s − 1.67·10-s − 0.377·11-s − 0.0335·12-s + 0.131·13-s + 0.411·14-s − 0.0641·15-s − 1.21·16-s − 0.459·17-s + 1.29·18-s + 1.16·19-s + 0.878·20-s + 0.0157·21-s + 0.488·22-s + 0.359·23-s − 0.0206·24-s + 0.678·25-s − 0.170·26-s + 0.0988·27-s − 0.215·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 - 1.72e10T \) |
| good | 2 | \( 1 + 234.T + 3.27e4T^{2} \) |
| 3 | \( 1 + 187.T + 1.43e7T^{2} \) |
| 5 | \( 1 - 2.26e5T + 3.05e10T^{2} \) |
| 7 | \( 1 + 6.91e5T + 4.74e12T^{2} \) |
| 11 | \( 1 + 2.43e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 2.97e7T + 5.11e16T^{2} \) |
| 17 | \( 1 + 7.77e8T + 2.86e18T^{2} \) |
| 19 | \( 1 - 4.53e9T + 1.51e19T^{2} \) |
| 23 | \( 1 - 5.86e9T + 2.66e20T^{2} \) |
| 31 | \( 1 - 1.20e11T + 2.34e22T^{2} \) |
| 37 | \( 1 + 4.34e9T + 3.33e23T^{2} \) |
| 41 | \( 1 + 9.99e11T + 1.55e24T^{2} \) |
| 43 | \( 1 + 7.15e11T + 3.17e24T^{2} \) |
| 47 | \( 1 + 2.21e12T + 1.20e25T^{2} \) |
| 53 | \( 1 + 9.43e12T + 7.31e25T^{2} \) |
| 59 | \( 1 + 8.43e12T + 3.65e26T^{2} \) |
| 61 | \( 1 - 2.35e13T + 6.02e26T^{2} \) |
| 67 | \( 1 - 3.43e13T + 2.46e27T^{2} \) |
| 71 | \( 1 - 5.75e13T + 5.87e27T^{2} \) |
| 73 | \( 1 + 4.90e13T + 8.90e27T^{2} \) |
| 79 | \( 1 + 2.68e14T + 2.91e28T^{2} \) |
| 83 | \( 1 + 2.87e14T + 6.11e28T^{2} \) |
| 89 | \( 1 + 1.80e14T + 1.74e29T^{2} \) |
| 97 | \( 1 - 1.62e14T + 6.33e29T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.25195078714719153137121277180, −11.37049662547573315541010367388, −10.12668095610313954479712041522, −9.326187892615432094229670075026, −8.208773802761398968979343994709, −6.59266645552973156969455000465, −5.23522451276662923913970465381, −2.75407516076682775695650092106, −1.40905910603059139466490752154, 0,
1.40905910603059139466490752154, 2.75407516076682775695650092106, 5.23522451276662923913970465381, 6.59266645552973156969455000465, 8.208773802761398968979343994709, 9.326187892615432094229670075026, 10.12668095610313954479712041522, 11.37049662547573315541010367388, 13.25195078714719153137121277180
