| L(s) = 1 | − 0.329·2-s − 3-s − 1.89·4-s − 2.73·5-s + 0.329·6-s − 7-s + 1.28·8-s + 9-s + 0.902·10-s − 2.50·11-s + 1.89·12-s + 1.48·13-s + 0.329·14-s + 2.73·15-s + 3.35·16-s + 0.902·17-s − 0.329·18-s + 2.50·19-s + 5.17·20-s + 21-s + 0.825·22-s + 23-s − 1.28·24-s + 2.48·25-s − 0.489·26-s − 27-s + 1.89·28-s + ⋯ |
| L(s) = 1 | − 0.233·2-s − 0.577·3-s − 0.945·4-s − 1.22·5-s + 0.134·6-s − 0.377·7-s + 0.453·8-s + 0.333·9-s + 0.285·10-s − 0.755·11-s + 0.545·12-s + 0.411·13-s + 0.0881·14-s + 0.706·15-s + 0.839·16-s + 0.218·17-s − 0.0777·18-s + 0.574·19-s + 1.15·20-s + 0.218·21-s + 0.176·22-s + 0.208·23-s − 0.261·24-s + 0.497·25-s − 0.0960·26-s − 0.192·27-s + 0.357·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5218855141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5218855141\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 0.329T + 2T^{2} \) |
| 5 | \( 1 + 2.73T + 5T^{2} \) |
| 11 | \( 1 + 2.50T + 11T^{2} \) |
| 13 | \( 1 - 1.48T + 13T^{2} \) |
| 17 | \( 1 - 0.902T + 17T^{2} \) |
| 19 | \( 1 - 2.50T + 19T^{2} \) |
| 29 | \( 1 - 6.68T + 29T^{2} \) |
| 31 | \( 1 - 1.09T + 31T^{2} \) |
| 37 | \( 1 - 9.91T + 37T^{2} \) |
| 41 | \( 1 + 2.30T + 41T^{2} \) |
| 43 | \( 1 - 5.83T + 43T^{2} \) |
| 47 | \( 1 + 6.74T + 47T^{2} \) |
| 53 | \( 1 + 4.91T + 53T^{2} \) |
| 59 | \( 1 + 7.32T + 59T^{2} \) |
| 61 | \( 1 + 1.00T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 + 0.362T + 71T^{2} \) |
| 73 | \( 1 + 5.34T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 - 7.59T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 13.9T + 97T^{2} \) |
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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
−10.96008943559054176619642983248, −10.12233743184832195710863243139, −9.229971071507852573500474793872, −8.112561881725781599125991155160, −7.64533888974531840341083698971, −6.33161498770041872464772859410, −5.12897949420093551423785093936, −4.29606734312514636847746370997, −3.22926501331519815345422729699, −0.70763662990958991157801470531,
0.70763662990958991157801470531, 3.22926501331519815345422729699, 4.29606734312514636847746370997, 5.12897949420093551423785093936, 6.33161498770041872464772859410, 7.64533888974531840341083698971, 8.112561881725781599125991155160, 9.229971071507852573500474793872, 10.12233743184832195710863243139, 10.96008943559054176619642983248
