| L(s) = 1 | − 2-s − 2.26·3-s + 4-s + 3.87·5-s + 2.26·6-s + 1.72·7-s − 8-s + 2.13·9-s − 3.87·10-s − 0.520·11-s − 2.26·12-s − 3.00·13-s − 1.72·14-s − 8.78·15-s + 16-s − 2.56·17-s − 2.13·18-s − 2.36·19-s + 3.87·20-s − 3.91·21-s + 0.520·22-s − 23-s + 2.26·24-s + 10.0·25-s + 3.00·26-s + 1.95·27-s + 1.72·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.30·3-s + 0.5·4-s + 1.73·5-s + 0.925·6-s + 0.652·7-s − 0.353·8-s + 0.712·9-s − 1.22·10-s − 0.156·11-s − 0.654·12-s − 0.834·13-s − 0.461·14-s − 2.26·15-s + 0.250·16-s − 0.621·17-s − 0.503·18-s − 0.543·19-s + 0.866·20-s − 0.854·21-s + 0.110·22-s − 0.208·23-s + 0.462·24-s + 2.00·25-s + 0.590·26-s + 0.376·27-s + 0.326·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 + 2.26T + 3T^{2} \) |
| 5 | \( 1 - 3.87T + 5T^{2} \) |
| 7 | \( 1 - 1.72T + 7T^{2} \) |
| 11 | \( 1 + 0.520T + 11T^{2} \) |
| 13 | \( 1 + 3.00T + 13T^{2} \) |
| 17 | \( 1 + 2.56T + 17T^{2} \) |
| 19 | \( 1 + 2.36T + 19T^{2} \) |
| 29 | \( 1 + 4.33T + 29T^{2} \) |
| 31 | \( 1 - 4.96T + 31T^{2} \) |
| 37 | \( 1 + 3.95T + 37T^{2} \) |
| 41 | \( 1 - 6.89T + 41T^{2} \) |
| 43 | \( 1 - 0.963T + 43T^{2} \) |
| 47 | \( 1 + 5.02T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 + 7.77T + 59T^{2} \) |
| 61 | \( 1 + 3.89T + 61T^{2} \) |
| 67 | \( 1 + 1.15T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 + 6.96T + 79T^{2} \) |
| 83 | \( 1 - 0.507T + 83T^{2} \) |
| 89 | \( 1 - 0.862T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−7.60010235022454790465404486357, −6.87984657978619911504924824158, −6.18329034956380129435583830553, −5.77508988079383602602040811112, −5.05297181559342955321703511577, −4.45988504895179407433396704271, −2.78059565361489251845495088328, −2.04171577226470200850764841567, −1.27574645118003034296692171154, 0,
1.27574645118003034296692171154, 2.04171577226470200850764841567, 2.78059565361489251845495088328, 4.45988504895179407433396704271, 5.05297181559342955321703511577, 5.77508988079383602602040811112, 6.18329034956380129435583830553, 6.87984657978619911504924824158, 7.60010235022454790465404486357
