| L(s) = 1 | + 1.35·2-s − 1.53·3-s − 0.157·4-s − 2.51·5-s − 2.08·6-s − 0.974·7-s − 2.92·8-s − 0.646·9-s − 3.41·10-s + 4.48·11-s + 0.241·12-s − 13-s − 1.32·14-s + 3.85·15-s − 3.66·16-s − 7.72·17-s − 0.878·18-s + 6.81·19-s + 0.395·20-s + 1.49·21-s + 6.09·22-s + 3.81·23-s + 4.49·24-s + 1.31·25-s − 1.35·26-s + 5.59·27-s + 0.153·28-s + ⋯ |
| L(s) = 1 | + 0.959·2-s − 0.885·3-s − 0.0786·4-s − 1.12·5-s − 0.850·6-s − 0.368·7-s − 1.03·8-s − 0.215·9-s − 1.07·10-s + 1.35·11-s + 0.0696·12-s − 0.277·13-s − 0.353·14-s + 0.995·15-s − 0.915·16-s − 1.87·17-s − 0.207·18-s + 1.56·19-s + 0.0884·20-s + 0.326·21-s + 1.29·22-s + 0.795·23-s + 0.916·24-s + 0.263·25-s − 0.266·26-s + 1.07·27-s + 0.0289·28-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
| \(L(1)\) |
\(\approx\) |
\(1.018180498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.018180498\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.35T + 2T^{2} \) |
| 3 | \( 1 + 1.53T + 3T^{2} \) |
| 5 | \( 1 + 2.51T + 5T^{2} \) |
| 7 | \( 1 + 0.974T + 7T^{2} \) |
| 11 | \( 1 - 4.48T + 11T^{2} \) |
| 17 | \( 1 + 7.72T + 17T^{2} \) |
| 19 | \( 1 - 6.81T + 19T^{2} \) |
| 23 | \( 1 - 3.81T + 23T^{2} \) |
| 29 | \( 1 - 1.02T + 29T^{2} \) |
| 31 | \( 1 - 9.51T + 31T^{2} \) |
| 37 | \( 1 + 3.87T + 37T^{2} \) |
| 41 | \( 1 - 7.89T + 41T^{2} \) |
| 43 | \( 1 + 4.19T + 43T^{2} \) |
| 47 | \( 1 - 4.88T + 47T^{2} \) |
| 53 | \( 1 - 1.99T + 53T^{2} \) |
| 59 | \( 1 + 3.95T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 + 4.14T + 67T^{2} \) |
| 71 | \( 1 - 8.40T + 71T^{2} \) |
| 73 | \( 1 - 15.4T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 14.0T + 89T^{2} \) |
| 97 | \( 1 + 3.08T + 97T^{2} \) |
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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
−9.419873059488033087578911360272, −8.971989528591018439266680163126, −7.905264148031743105561359325658, −6.64239051362888775903877521061, −6.42107915986298423334993102146, −5.18907942703319460684202340554, −4.56336452388066197740575280781, −3.77727733593955543816643981466, −2.86928981908916389110670510933, −0.65301908965811891339066456154,
0.65301908965811891339066456154, 2.86928981908916389110670510933, 3.77727733593955543816643981466, 4.56336452388066197740575280781, 5.18907942703319460684202340554, 6.42107915986298423334993102146, 6.64239051362888775903877521061, 7.905264148031743105561359325658, 8.971989528591018439266680163126, 9.419873059488033087578911360272
