| L(s) = 1 | − 2-s − 3-s + 4-s + 3.10·5-s + 6-s + 0.399·7-s − 8-s + 9-s − 3.10·10-s + 5.91·11-s − 12-s + 3.54·13-s − 0.399·14-s − 3.10·15-s + 16-s − 17-s − 18-s − 1.83·19-s + 3.10·20-s − 0.399·21-s − 5.91·22-s − 1.17·23-s + 24-s + 4.66·25-s − 3.54·26-s − 27-s + 0.399·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.39·5-s + 0.408·6-s + 0.151·7-s − 0.353·8-s + 0.333·9-s − 0.982·10-s + 1.78·11-s − 0.288·12-s + 0.984·13-s − 0.106·14-s − 0.802·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s − 0.421·19-s + 0.695·20-s − 0.0872·21-s − 1.26·22-s − 0.245·23-s + 0.204·24-s + 0.932·25-s − 0.696·26-s − 0.192·27-s + 0.0755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.994664328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.994664328\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 5 | \( 1 - 3.10T + 5T^{2} \) |
| 7 | \( 1 - 0.399T + 7T^{2} \) |
| 11 | \( 1 - 5.91T + 11T^{2} \) |
| 13 | \( 1 - 3.54T + 13T^{2} \) |
| 19 | \( 1 + 1.83T + 19T^{2} \) |
| 23 | \( 1 + 1.17T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 - 0.942T + 31T^{2} \) |
| 37 | \( 1 - 5.82T + 37T^{2} \) |
| 41 | \( 1 - 0.225T + 41T^{2} \) |
| 43 | \( 1 - 6.42T + 43T^{2} \) |
| 47 | \( 1 - 7.48T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 61 | \( 1 + 1.23T + 61T^{2} \) |
| 67 | \( 1 + 7.88T + 67T^{2} \) |
| 71 | \( 1 - 6.19T + 71T^{2} \) |
| 73 | \( 1 + 8.51T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 - 7.33T + 83T^{2} \) |
| 89 | \( 1 + 5.20T + 89T^{2} \) |
| 97 | \( 1 + 1.51T + 97T^{2} \) |
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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
−8.240677182227602166667163202806, −7.20524101052417126569003911608, −6.54044113588584652660218093911, −6.05026438374481186480528388826, −5.61540285406971371332415761227, −4.41428327474238613318254936309, −3.70826845321722116980687816509, −2.40238664839854412918965908282, −1.60390380624061051200791625532, −0.939954484996958543302151157345,
0.939954484996958543302151157345, 1.60390380624061051200791625532, 2.40238664839854412918965908282, 3.70826845321722116980687816509, 4.41428327474238613318254936309, 5.61540285406971371332415761227, 6.05026438374481186480528388826, 6.54044113588584652660218093911, 7.20524101052417126569003911608, 8.240677182227602166667163202806
