| L(s) = 1 | + 0.665·2-s − 3.11·3-s − 1.55·4-s − 0.365·5-s − 2.07·6-s − 1.41·7-s − 2.36·8-s + 6.71·9-s − 0.243·10-s + 11-s + 4.85·12-s + 2.19·13-s − 0.944·14-s + 1.14·15-s + 1.53·16-s + 7.23·17-s + 4.46·18-s − 2.16·19-s + 0.569·20-s + 4.42·21-s + 0.665·22-s + 2.10·23-s + 7.37·24-s − 4.86·25-s + 1.45·26-s − 11.5·27-s + 2.21·28-s + ⋯ |
| L(s) = 1 | + 0.470·2-s − 1.79·3-s − 0.778·4-s − 0.163·5-s − 0.846·6-s − 0.536·7-s − 0.836·8-s + 2.23·9-s − 0.0770·10-s + 0.301·11-s + 1.40·12-s + 0.608·13-s − 0.252·14-s + 0.294·15-s + 0.384·16-s + 1.75·17-s + 1.05·18-s − 0.497·19-s + 0.127·20-s + 0.965·21-s + 0.141·22-s + 0.439·23-s + 1.50·24-s − 0.973·25-s + 0.286·26-s − 2.22·27-s + 0.417·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 2 | \( 1 - 0.665T + 2T^{2} \) |
| 3 | \( 1 + 3.11T + 3T^{2} \) |
| 5 | \( 1 + 0.365T + 5T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 13 | \( 1 - 2.19T + 13T^{2} \) |
| 17 | \( 1 - 7.23T + 17T^{2} \) |
| 19 | \( 1 + 2.16T + 19T^{2} \) |
| 23 | \( 1 - 2.10T + 23T^{2} \) |
| 29 | \( 1 - 1.53T + 29T^{2} \) |
| 31 | \( 1 + 6.46T + 31T^{2} \) |
| 37 | \( 1 + 5.05T + 37T^{2} \) |
| 41 | \( 1 - 3.54T + 41T^{2} \) |
| 43 | \( 1 - 3.24T + 43T^{2} \) |
| 47 | \( 1 + 9.24T + 47T^{2} \) |
| 53 | \( 1 - 4.01T + 53T^{2} \) |
| 59 | \( 1 + 1.83T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 + 7.96T + 67T^{2} \) |
| 71 | \( 1 - 0.988T + 71T^{2} \) |
| 73 | \( 1 - 0.0955T + 73T^{2} \) |
| 79 | \( 1 + 9.70T + 79T^{2} \) |
| 83 | \( 1 - 9.43T + 83T^{2} \) |
| 89 | \( 1 + 5.27T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−9.428764663168508062158511279441, −8.285661065595731268443367069454, −7.23679686885326622425015053238, −6.31335881467075602082940136521, −5.70638045539072137797043939417, −5.14139003771224399749336576835, −4.12341162096713079988218152029, −3.43286034770348742221721308366, −1.23550367969478907518531935475, 0,
1.23550367969478907518531935475, 3.43286034770348742221721308366, 4.12341162096713079988218152029, 5.14139003771224399749336576835, 5.70638045539072137797043939417, 6.31335881467075602082940136521, 7.23679686885326622425015053238, 8.285661065595731268443367069454, 9.428764663168508062158511279441
