| L(s) = 1 | + 2.20·2-s + 2.69·3-s + 2.86·4-s + 0.920·5-s + 5.93·6-s − 2.62·7-s + 1.90·8-s + 4.23·9-s + 2.02·10-s − 3.87·11-s + 7.70·12-s + 0.665·13-s − 5.78·14-s + 2.47·15-s − 1.52·16-s + 17-s + 9.34·18-s + 1.11·19-s + 2.63·20-s − 7.05·21-s − 8.53·22-s + 9.12·23-s + 5.13·24-s − 4.15·25-s + 1.46·26-s + 3.33·27-s − 7.51·28-s + ⋯ |
| L(s) = 1 | + 1.55·2-s + 1.55·3-s + 1.43·4-s + 0.411·5-s + 2.42·6-s − 0.991·7-s + 0.674·8-s + 1.41·9-s + 0.641·10-s − 1.16·11-s + 2.22·12-s + 0.184·13-s − 1.54·14-s + 0.639·15-s − 0.380·16-s + 0.242·17-s + 2.20·18-s + 0.256·19-s + 0.589·20-s − 1.53·21-s − 1.81·22-s + 1.90·23-s + 1.04·24-s − 0.830·25-s + 0.287·26-s + 0.641·27-s − 1.42·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.137850379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.137850379\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 - 2.20T + 2T^{2} \) |
| 3 | \( 1 - 2.69T + 3T^{2} \) |
| 5 | \( 1 - 0.920T + 5T^{2} \) |
| 7 | \( 1 + 2.62T + 7T^{2} \) |
| 11 | \( 1 + 3.87T + 11T^{2} \) |
| 13 | \( 1 - 0.665T + 13T^{2} \) |
| 19 | \( 1 - 1.11T + 19T^{2} \) |
| 23 | \( 1 - 9.12T + 23T^{2} \) |
| 29 | \( 1 + 3.52T + 29T^{2} \) |
| 31 | \( 1 - 4.39T + 31T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 - 7.18T + 41T^{2} \) |
| 47 | \( 1 + 4.91T + 47T^{2} \) |
| 53 | \( 1 - 2.14T + 53T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 - 14.3T + 61T^{2} \) |
| 67 | \( 1 - 9.51T + 67T^{2} \) |
| 71 | \( 1 + 6.27T + 71T^{2} \) |
| 73 | \( 1 - 3.62T + 73T^{2} \) |
| 79 | \( 1 + 6.08T + 79T^{2} \) |
| 83 | \( 1 - 5.14T + 83T^{2} \) |
| 89 | \( 1 - 2.88T + 89T^{2} \) |
| 97 | \( 1 - 3.05T + 97T^{2} \) |
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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
−10.31152539301858959059820224433, −9.490776658531579626613072440941, −8.724663965503619484745827587780, −7.61196393361855546117604746238, −6.80063476425251718391936022444, −5.73577858318939108526065087863, −4.83652940432844744011697836559, −3.53893059335847929221127727658, −3.08141482526327911133473497842, −2.19095600289254623169227932908,
2.19095600289254623169227932908, 3.08141482526327911133473497842, 3.53893059335847929221127727658, 4.83652940432844744011697836559, 5.73577858318939108526065087863, 6.80063476425251718391936022444, 7.61196393361855546117604746238, 8.724663965503619484745827587780, 9.490776658531579626613072440941, 10.31152539301858959059820224433
