| L(s) = 1 | + 2·2-s − 4·3-s + 4·4-s + 14.5·5-s − 8·6-s − 7·7-s + 8·8-s − 11·9-s + 29.1·10-s + 45.1·11-s − 16·12-s − 14·14-s − 58.3·15-s + 16·16-s − 71.1·17-s − 22·18-s − 140.·19-s + 58.3·20-s + 28·21-s + 90.3·22-s + 35.4·23-s − 32·24-s + 88.0·25-s + 152·27-s − 28·28-s + 153.·29-s − 116.·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.769·3-s + 0.5·4-s + 1.30·5-s − 0.544·6-s − 0.377·7-s + 0.353·8-s − 0.407·9-s + 0.923·10-s + 1.23·11-s − 0.384·12-s − 0.267·14-s − 1.00·15-s + 0.250·16-s − 1.01·17-s − 0.288·18-s − 1.69·19-s + 0.652·20-s + 0.290·21-s + 0.875·22-s + 0.320·23-s − 0.272·24-s + 0.704·25-s + 1.08·27-s − 0.188·28-s + 0.982·29-s − 0.710·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 7 | \( 1 + 7T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 4T + 27T^{2} \) |
| 5 | \( 1 - 14.5T + 125T^{2} \) |
| 11 | \( 1 - 45.1T + 1.33e3T^{2} \) |
| 17 | \( 1 + 71.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 140.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 35.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 153.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 113.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 223.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 404.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 30.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 328.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 362.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 236.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 493.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 293.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.07e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 752.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.00e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 548.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 694.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 5.98T + 9.12e5T^{2} \) |
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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
−8.476088552085037745573926499430, −6.85839810251648958785461763935, −6.34405439342038132053796092821, −6.17478075751049121811962919268, −5.06742963333999857263488454760, −4.47982527325111581552689113585, −3.31324294067272613993237208842, −2.28878782595099554496952072558, −1.44061258118458080062693752912, 0,
1.44061258118458080062693752912, 2.28878782595099554496952072558, 3.31324294067272613993237208842, 4.47982527325111581552689113585, 5.06742963333999857263488454760, 6.17478075751049121811962919268, 6.34405439342038132053796092821, 6.85839810251648958785461763935, 8.476088552085037745573926499430
