| L(s) = 1 | − 2.17·2-s − 2.70·3-s + 2.70·4-s − 4.17·5-s + 5.87·6-s − 1.70·7-s − 1.53·8-s + 4.34·9-s + 9.04·10-s − 4.63·11-s − 7.34·12-s + 13-s + 3.70·14-s + 11.2·15-s − 2.07·16-s − 4.41·17-s − 9.41·18-s − 4.80·19-s − 11.2·20-s + 4.63·21-s + 10.0·22-s + 7.60·23-s + 4.17·24-s + 12.3·25-s − 2.17·26-s − 3.63·27-s − 4.63·28-s + ⋯ |
| L(s) = 1 | − 1.53·2-s − 1.56·3-s + 1.35·4-s − 1.86·5-s + 2.40·6-s − 0.646·7-s − 0.544·8-s + 1.44·9-s + 2.86·10-s − 1.39·11-s − 2.11·12-s + 0.277·13-s + 0.991·14-s + 2.91·15-s − 0.519·16-s − 1.07·17-s − 2.21·18-s − 1.10·19-s − 2.52·20-s + 1.01·21-s + 2.14·22-s + 1.58·23-s + 0.851·24-s + 2.47·25-s − 0.425·26-s − 0.698·27-s − 0.875·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 2.17T + 2T^{2} \) |
| 3 | \( 1 + 2.70T + 3T^{2} \) |
| 5 | \( 1 + 4.17T + 5T^{2} \) |
| 7 | \( 1 + 1.70T + 7T^{2} \) |
| 11 | \( 1 + 4.63T + 11T^{2} \) |
| 17 | \( 1 + 4.41T + 17T^{2} \) |
| 19 | \( 1 + 4.80T + 19T^{2} \) |
| 23 | \( 1 - 7.60T + 23T^{2} \) |
| 29 | \( 1 + 3.29T + 29T^{2} \) |
| 31 | \( 1 - 5.75T + 31T^{2} \) |
| 37 | \( 1 - 7.58T + 37T^{2} \) |
| 41 | \( 1 + 5.26T + 41T^{2} \) |
| 43 | \( 1 - 6.34T + 43T^{2} \) |
| 47 | \( 1 - 8.38T + 47T^{2} \) |
| 53 | \( 1 - 2.04T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + 4.68T + 67T^{2} \) |
| 71 | \( 1 - 8.52T + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 - 0.326T + 83T^{2} \) |
| 89 | \( 1 - 1.06T + 89T^{2} \) |
| 97 | \( 1 - 4.73T + 97T^{2} \) |
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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
−9.120717649825278668788871328873, −8.378618604513700836006543010479, −7.61455349790650528441585672606, −6.96607345943087411242264493281, −6.28657038268814874053185084767, −4.93597821966455124410520437796, −4.23890140503091480871358111845, −2.73700805129373733884572911404, −0.73749877297560804180048932186, 0,
0.73749877297560804180048932186, 2.73700805129373733884572911404, 4.23890140503091480871358111845, 4.93597821966455124410520437796, 6.28657038268814874053185084767, 6.96607345943087411242264493281, 7.61455349790650528441585672606, 8.378618604513700836006543010479, 9.120717649825278668788871328873
