| L(s) = 1 | − 0.498·2-s − 0.240·3-s − 1.75·4-s + 0.581·5-s + 0.120·6-s + 1.41·7-s + 1.87·8-s − 2.94·9-s − 0.290·10-s − 11-s + 0.421·12-s − 0.703·14-s − 0.139·15-s + 2.56·16-s + 0.451·17-s + 1.46·18-s + 3.19·19-s − 1.01·20-s − 0.339·21-s + 0.498·22-s − 4.03·23-s − 0.450·24-s − 4.66·25-s + 1.42·27-s − 2.46·28-s + 10.2·29-s + 0.0698·30-s + ⋯ |
| L(s) = 1 | − 0.352·2-s − 0.138·3-s − 0.875·4-s + 0.260·5-s + 0.0490·6-s + 0.532·7-s + 0.661·8-s − 0.980·9-s − 0.0917·10-s − 0.301·11-s + 0.121·12-s − 0.188·14-s − 0.0361·15-s + 0.642·16-s + 0.109·17-s + 0.346·18-s + 0.731·19-s − 0.227·20-s − 0.0740·21-s + 0.106·22-s − 0.841·23-s − 0.0919·24-s − 0.932·25-s + 0.275·27-s − 0.466·28-s + 1.89·29-s + 0.0127·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.498T + 2T^{2} \) |
| 3 | \( 1 + 0.240T + 3T^{2} \) |
| 5 | \( 1 - 0.581T + 5T^{2} \) |
| 7 | \( 1 - 1.41T + 7T^{2} \) |
| 17 | \( 1 - 0.451T + 17T^{2} \) |
| 19 | \( 1 - 3.19T + 19T^{2} \) |
| 23 | \( 1 + 4.03T + 23T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 + 7.10T + 31T^{2} \) |
| 37 | \( 1 - 4.68T + 37T^{2} \) |
| 41 | \( 1 + 8.86T + 41T^{2} \) |
| 43 | \( 1 - 4.41T + 43T^{2} \) |
| 47 | \( 1 - 3.77T + 47T^{2} \) |
| 53 | \( 1 + 0.0779T + 53T^{2} \) |
| 59 | \( 1 + 7.70T + 59T^{2} \) |
| 61 | \( 1 + 4.32T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 + 1.98T + 71T^{2} \) |
| 73 | \( 1 + 2.84T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 - 4.03T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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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
−8.860518430387900737396966826060, −8.084200653378351353704628093556, −7.64527406075150101658404105120, −6.31812783637604135308094167119, −5.50750683738869252064361434447, −4.88784507335579459977639407730, −3.89381949947417730949640311926, −2.77411930523072135462802720201, −1.43736190876445927494243573855, 0,
1.43736190876445927494243573855, 2.77411930523072135462802720201, 3.89381949947417730949640311926, 4.88784507335579459977639407730, 5.50750683738869252064361434447, 6.31812783637604135308094167119, 7.64527406075150101658404105120, 8.084200653378351353704628093556, 8.860518430387900737396966826060
