| L(s) = 1 | + 1.73·2-s + 0.999·4-s − 1.73·5-s − 1.73·8-s − 2.99·10-s − 5·16-s − 3·17-s + 3.46·19-s − 1.73·20-s − 6·23-s − 2.00·25-s − 3·29-s − 3.46·31-s − 5.19·32-s − 5.19·34-s − 8.66·37-s + 5.99·38-s + 3.00·40-s + 5.19·41-s − 8·43-s − 10.3·46-s + 3.46·47-s − 7·49-s − 3.46·50-s + 3·53-s − 5.19·58-s − 6.92·59-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.499·4-s − 0.774·5-s − 0.612·8-s − 0.948·10-s − 1.25·16-s − 0.727·17-s + 0.794·19-s − 0.387·20-s − 1.25·23-s − 0.400·25-s − 0.557·29-s − 0.622·31-s − 0.918·32-s − 0.891·34-s − 1.42·37-s + 0.973·38-s + 0.474·40-s + 0.811·41-s − 1.21·43-s − 1.53·46-s + 0.505·47-s − 49-s − 0.489·50-s + 0.412·53-s − 0.682·58-s − 0.901·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 5 | \( 1 + 1.73T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 + 8.66T + 37T^{2} \) |
| 41 | \( 1 - 5.19T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 - 3.46T + 67T^{2} \) |
| 71 | \( 1 - 3.46T + 71T^{2} \) |
| 73 | \( 1 - 1.73T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 6.92T + 89T^{2} \) |
| 97 | \( 1 + 6.92T + 97T^{2} \) |
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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
−9.041723559381136267824340917759, −8.163524128386339054488294127688, −7.33113071713810766905902237075, −6.44118590881199747061221580175, −5.58788361213582033437527275522, −4.79260304176934536619508344813, −3.91304747946918563289983767918, −3.36480744552396524026032289560, −2.07269884376571401821555132095, 0,
2.07269884376571401821555132095, 3.36480744552396524026032289560, 3.91304747946918563289983767918, 4.79260304176934536619508344813, 5.58788361213582033437527275522, 6.44118590881199747061221580175, 7.33113071713810766905902237075, 8.163524128386339054488294127688, 9.041723559381136267824340917759
