| L(s) = 1 | + 2.75·2-s − 23.9·3-s − 24.4·4-s − 66.0·6-s − 85.7·7-s − 155.·8-s + 331.·9-s + 431.·11-s + 584.·12-s − 169·13-s − 236.·14-s + 352.·16-s + 438.·17-s + 912.·18-s + 1.61e3·19-s + 2.05e3·21-s + 1.18e3·22-s − 2.17e3·23-s + 3.72e3·24-s − 465.·26-s − 2.11e3·27-s + 2.09e3·28-s + 8.28e3·29-s − 2.74e3·31-s + 5.94e3·32-s − 1.03e4·33-s + 1.20e3·34-s + ⋯ |
| L(s) = 1 | + 0.487·2-s − 1.53·3-s − 0.762·4-s − 0.749·6-s − 0.661·7-s − 0.858·8-s + 1.36·9-s + 1.07·11-s + 1.17·12-s − 0.277·13-s − 0.322·14-s + 0.343·16-s + 0.368·17-s + 0.664·18-s + 1.02·19-s + 1.01·21-s + 0.523·22-s − 0.856·23-s + 1.32·24-s − 0.135·26-s − 0.557·27-s + 0.504·28-s + 1.83·29-s − 0.513·31-s + 1.02·32-s − 1.65·33-s + 0.179·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 + 169T \) |
| good | 2 | \( 1 - 2.75T + 32T^{2} \) |
| 3 | \( 1 + 23.9T + 243T^{2} \) |
| 7 | \( 1 + 85.7T + 1.68e4T^{2} \) |
| 11 | \( 1 - 431.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 438.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.61e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.17e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.28e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.74e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.13e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.95e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.15e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.32e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.75e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.97e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.55e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.94e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.42e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.02e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.07e5T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.64e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.41e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.33e5T + 8.58e9T^{2} \) |
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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
−10.21350646656407414254254783013, −9.707675641774025140304302535021, −8.513772443252950137087218591348, −6.95908850932545750520435339210, −6.16036106965174051724777938539, −5.34634751750776506845171525647, −4.43428335238129209060214704046, −3.33289960411406634424279978995, −1.07708091498980695236644692499, 0,
1.07708091498980695236644692499, 3.33289960411406634424279978995, 4.43428335238129209060214704046, 5.34634751750776506845171525647, 6.16036106965174051724777938539, 6.95908850932545750520435339210, 8.513772443252950137087218591348, 9.707675641774025140304302535021, 10.21350646656407414254254783013
