| L(s) = 1 | − 9.27·2-s + 53.9·4-s + 37.4·5-s + 154.·7-s − 203.·8-s − 347.·10-s − 521.·11-s − 28.8·13-s − 1.43e3·14-s + 163.·16-s + 428.·17-s − 2.56e3·19-s + 2.02e3·20-s + 4.83e3·22-s + 529·23-s − 1.71e3·25-s + 267.·26-s + 8.36e3·28-s + 2.40e3·29-s − 2.13e3·31-s + 5.01e3·32-s − 3.97e3·34-s + 5.80e3·35-s + 3.65e3·37-s + 2.37e4·38-s − 7.64e3·40-s − 1.61e4·41-s + ⋯ |
| L(s) = 1 | − 1.63·2-s + 1.68·4-s + 0.670·5-s + 1.19·7-s − 1.12·8-s − 1.09·10-s − 1.29·11-s − 0.0473·13-s − 1.95·14-s + 0.159·16-s + 0.359·17-s − 1.62·19-s + 1.13·20-s + 2.12·22-s + 0.208·23-s − 0.550·25-s + 0.0776·26-s + 2.01·28-s + 0.530·29-s − 0.398·31-s + 0.865·32-s − 0.589·34-s + 0.801·35-s + 0.438·37-s + 2.67·38-s − 0.755·40-s − 1.50·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 - 529T \) |
| good | 2 | \( 1 + 9.27T + 32T^{2} \) |
| 5 | \( 1 - 37.4T + 3.12e3T^{2} \) |
| 7 | \( 1 - 154.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 521.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 28.8T + 3.71e5T^{2} \) |
| 17 | \( 1 - 428.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.56e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 2.40e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.13e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.65e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.61e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.50e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.07e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.43e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.82e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.65e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.63e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.94e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.74e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.73e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.69e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.29e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.14e4T + 8.58e9T^{2} \) |
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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
−10.62638777053200477302341720005, −10.14489458231613695467131786530, −8.926446015330390988174634306322, −8.170593686539017073621629407696, −7.41884863581944734008160860194, −6.04633231004228032126938975173, −4.74863620530129369541939914960, −2.43922667383458701271959818125, −1.53835021038652935217207778691, 0,
1.53835021038652935217207778691, 2.43922667383458701271959818125, 4.74863620530129369541939914960, 6.04633231004228032126938975173, 7.41884863581944734008160860194, 8.170593686539017073621629407696, 8.926446015330390988174634306322, 10.14489458231613695467131786530, 10.62638777053200477302341720005
