L(s) = 1 | − 0.869·2-s + 2.57·3-s − 1.24·4-s − 3.57·5-s − 2.24·6-s + 2.82·8-s + 3.64·9-s + 3.11·10-s + 1.73·11-s − 3.20·12-s − 2.93·13-s − 9.22·15-s + 0.0316·16-s + 17-s − 3.17·18-s − 2.80·19-s + 4.44·20-s − 1.51·22-s − 7.96·23-s + 7.27·24-s + 7.80·25-s + 2.54·26-s + 1.67·27-s + 1.73·29-s + 8.02·30-s − 10.0·31-s − 5.67·32-s + ⋯ |
L(s) = 1 | − 0.615·2-s + 1.48·3-s − 0.621·4-s − 1.60·5-s − 0.915·6-s + 0.997·8-s + 1.21·9-s + 0.984·10-s + 0.524·11-s − 0.925·12-s − 0.812·13-s − 2.38·15-s + 0.00790·16-s + 0.242·17-s − 0.748·18-s − 0.644·19-s + 0.994·20-s − 0.322·22-s − 1.66·23-s + 1.48·24-s + 1.56·25-s + 0.499·26-s + 0.321·27-s + 0.323·29-s + 1.46·30-s − 1.80·31-s − 1.00·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 2 | \( 1 + 0.869T + 2T^{2} \) |
| 3 | \( 1 - 2.57T + 3T^{2} \) |
| 5 | \( 1 + 3.57T + 5T^{2} \) |
| 11 | \( 1 - 1.73T + 11T^{2} \) |
| 13 | \( 1 + 2.93T + 13T^{2} \) |
| 19 | \( 1 + 2.80T + 19T^{2} \) |
| 23 | \( 1 + 7.96T + 23T^{2} \) |
| 29 | \( 1 - 1.73T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 1.19T + 37T^{2} \) |
| 41 | \( 1 + 4.83T + 41T^{2} \) |
| 43 | \( 1 + 12.5T + 43T^{2} \) |
| 47 | \( 1 + 3.55T + 47T^{2} \) |
| 53 | \( 1 + 8.64T + 53T^{2} \) |
| 59 | \( 1 - 8.97T + 59T^{2} \) |
| 61 | \( 1 - 3.79T + 61T^{2} \) |
| 67 | \( 1 - 3.13T + 67T^{2} \) |
| 71 | \( 1 - 9.12T + 71T^{2} \) |
| 73 | \( 1 + 3.16T + 73T^{2} \) |
| 79 | \( 1 + 0.444T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 - 17.8T + 89T^{2} \) |
| 97 | \( 1 - 2.73T + 97T^{2} \) |
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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
−9.558451997930711461332909857646, −8.778635850202821605832324695128, −8.086262763852716886259605281108, −7.78999056453534355154760523433, −6.84159894814175646392850486814, −4.94470898396763776829715236624, −3.93977498301663370885167733901, −3.51066487602760177475928264341, −1.93294958212395504423073755238, 0,
1.93294958212395504423073755238, 3.51066487602760177475928264341, 3.93977498301663370885167733901, 4.94470898396763776829715236624, 6.84159894814175646392850486814, 7.78999056453534355154760523433, 8.086262763852716886259605281108, 8.778635850202821605832324695128, 9.558451997930711461332909857646