| L(s) = 1 | + 2.79·2-s + 3-s + 5.78·4-s + 1.94·5-s + 2.79·6-s − 1.90·7-s + 10.5·8-s + 9-s + 5.42·10-s + 0.728·11-s + 5.78·12-s − 5.07·13-s − 5.32·14-s + 1.94·15-s + 17.9·16-s + 17-s + 2.79·18-s + 2.91·19-s + 11.2·20-s − 1.90·21-s + 2.03·22-s − 1.70·23-s + 10.5·24-s − 1.22·25-s − 14.1·26-s + 27-s − 11.0·28-s + ⋯ |
| L(s) = 1 | + 1.97·2-s + 0.577·3-s + 2.89·4-s + 0.868·5-s + 1.13·6-s − 0.721·7-s + 3.73·8-s + 0.333·9-s + 1.71·10-s + 0.219·11-s + 1.67·12-s − 1.40·13-s − 1.42·14-s + 0.501·15-s + 4.48·16-s + 0.242·17-s + 0.657·18-s + 0.667·19-s + 2.51·20-s − 0.416·21-s + 0.433·22-s − 0.355·23-s + 2.15·24-s − 0.245·25-s − 2.77·26-s + 0.192·27-s − 2.08·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.747096921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.747096921\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 - 2.79T + 2T^{2} \) |
| 5 | \( 1 - 1.94T + 5T^{2} \) |
| 7 | \( 1 + 1.90T + 7T^{2} \) |
| 11 | \( 1 - 0.728T + 11T^{2} \) |
| 13 | \( 1 + 5.07T + 13T^{2} \) |
| 19 | \( 1 - 2.91T + 19T^{2} \) |
| 23 | \( 1 + 1.70T + 23T^{2} \) |
| 29 | \( 1 - 1.56T + 29T^{2} \) |
| 31 | \( 1 - 3.84T + 31T^{2} \) |
| 37 | \( 1 - 5.97T + 37T^{2} \) |
| 41 | \( 1 + 6.77T + 41T^{2} \) |
| 43 | \( 1 + 8.00T + 43T^{2} \) |
| 47 | \( 1 + 3.49T + 47T^{2} \) |
| 53 | \( 1 + 0.650T + 53T^{2} \) |
| 59 | \( 1 - 4.88T + 59T^{2} \) |
| 61 | \( 1 + 4.59T + 61T^{2} \) |
| 67 | \( 1 + 7.38T + 67T^{2} \) |
| 71 | \( 1 + 2.04T + 71T^{2} \) |
| 73 | \( 1 - 6.82T + 73T^{2} \) |
| 83 | \( 1 - 0.725T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 - 1.34T + 97T^{2} \) |
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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
−8.076977607721381956663774029478, −7.42136878539698036302474106955, −6.61680431779893495062882717052, −6.20291392018517120185733860497, −5.26483140794422823914939845013, −4.77389260977519388469662382865, −3.79535923535266586148588491752, −3.03590039567320852183718432402, −2.44736595893419379557230168936, −1.57699285983741823903322209226,
1.57699285983741823903322209226, 2.44736595893419379557230168936, 3.03590039567320852183718432402, 3.79535923535266586148588491752, 4.77389260977519388469662382865, 5.26483140794422823914939845013, 6.20291392018517120185733860497, 6.61680431779893495062882717052, 7.42136878539698036302474106955, 8.076977607721381956663774029478
