| L(s) = 1 | + 1.22·2-s + 2.22·3-s − 0.495·4-s − 2.88·5-s + 2.72·6-s + 1.14·7-s − 3.06·8-s + 1.95·9-s − 3.54·10-s − 3.96·11-s − 1.10·12-s − 1.63·13-s + 1.40·14-s − 6.42·15-s − 2.76·16-s + 1.55·17-s + 2.39·18-s − 3.44·19-s + 1.43·20-s + 2.55·21-s − 4.85·22-s − 2.07·23-s − 6.81·24-s + 3.34·25-s − 2.00·26-s − 2.32·27-s − 0.568·28-s + ⋯ |
| L(s) = 1 | + 0.867·2-s + 1.28·3-s − 0.247·4-s − 1.29·5-s + 1.11·6-s + 0.433·7-s − 1.08·8-s + 0.651·9-s − 1.12·10-s − 1.19·11-s − 0.318·12-s − 0.452·13-s + 0.375·14-s − 1.66·15-s − 0.690·16-s + 0.378·17-s + 0.564·18-s − 0.790·19-s + 0.320·20-s + 0.557·21-s − 1.03·22-s − 0.433·23-s − 1.39·24-s + 0.669·25-s − 0.392·26-s − 0.447·27-s − 0.107·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1511 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 1511 | \( 1 + T \) |
| good | 2 | \( 1 - 1.22T + 2T^{2} \) |
| 3 | \( 1 - 2.22T + 3T^{2} \) |
| 5 | \( 1 + 2.88T + 5T^{2} \) |
| 7 | \( 1 - 1.14T + 7T^{2} \) |
| 11 | \( 1 + 3.96T + 11T^{2} \) |
| 13 | \( 1 + 1.63T + 13T^{2} \) |
| 17 | \( 1 - 1.55T + 17T^{2} \) |
| 19 | \( 1 + 3.44T + 19T^{2} \) |
| 23 | \( 1 + 2.07T + 23T^{2} \) |
| 29 | \( 1 - 3.30T + 29T^{2} \) |
| 31 | \( 1 + 0.616T + 31T^{2} \) |
| 37 | \( 1 + 7.26T + 37T^{2} \) |
| 41 | \( 1 + 4.05T + 41T^{2} \) |
| 43 | \( 1 - 0.398T + 43T^{2} \) |
| 47 | \( 1 + 1.30T + 47T^{2} \) |
| 53 | \( 1 + 0.123T + 53T^{2} \) |
| 59 | \( 1 - 4.59T + 59T^{2} \) |
| 61 | \( 1 + 4.31T + 61T^{2} \) |
| 67 | \( 1 + 0.417T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 + 3.33T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + 1.80T + 97T^{2} \) |
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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
−8.798631590615749191820594277273, −8.108320971079341851014846468343, −7.87292292999425471875827337408, −6.74756603768561537791762415932, −5.40251007427184553000316111928, −4.68206872260620084134978677843, −3.83237305527161203411190895180, −3.19918811650398302169591298912, −2.27047423114845849381443397244, 0,
2.27047423114845849381443397244, 3.19918811650398302169591298912, 3.83237305527161203411190895180, 4.68206872260620084134978677843, 5.40251007427184553000316111928, 6.74756603768561537791762415932, 7.87292292999425471875827337408, 8.108320971079341851014846468343, 8.798631590615749191820594277273
