| L(s) = 1 | + 4.75·2-s − 3·3-s + 14.6·4-s − 3.92·5-s − 14.2·6-s − 7·7-s + 31.6·8-s + 9·9-s − 18.6·10-s − 72.4·11-s − 43.9·12-s + 2.47·13-s − 33.3·14-s + 11.7·15-s + 33.2·16-s + 20.3·17-s + 42.8·18-s + 1.34·19-s − 57.4·20-s + 21·21-s − 344.·22-s + 23·23-s − 94.8·24-s − 109.·25-s + 11.7·26-s − 27·27-s − 102.·28-s + ⋯ |
| L(s) = 1 | + 1.68·2-s − 0.577·3-s + 1.83·4-s − 0.351·5-s − 0.971·6-s − 0.377·7-s + 1.39·8-s + 0.333·9-s − 0.590·10-s − 1.98·11-s − 1.05·12-s + 0.0528·13-s − 0.635·14-s + 0.202·15-s + 0.519·16-s + 0.290·17-s + 0.560·18-s + 0.0162·19-s − 0.642·20-s + 0.218·21-s − 3.34·22-s + 0.208·23-s − 0.806·24-s − 0.876·25-s + 0.0889·26-s − 0.192·27-s − 0.691·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 - 4.75T + 8T^{2} \) |
| 5 | \( 1 + 3.92T + 125T^{2} \) |
| 11 | \( 1 + 72.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 2.47T + 2.19e3T^{2} \) |
| 17 | \( 1 - 20.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 1.34T + 6.85e3T^{2} \) |
| 29 | \( 1 + 214.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 236.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 6.88T + 5.06e4T^{2} \) |
| 41 | \( 1 - 327.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 77.6T + 7.95e4T^{2} \) |
| 47 | \( 1 - 559.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 584.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 413.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 876.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 908.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 471.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 657.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 852.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 11.1T + 5.71e5T^{2} \) |
| 89 | \( 1 - 186.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.29e3T + 9.12e5T^{2} \) |
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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
−10.69417132505853747673508336393, −9.438673877808763745213372322799, −7.79737690319266234323973828432, −7.17903420722688375819191659125, −5.81303700718202349463408303074, −5.47783897817210982198422627307, −4.38188688767321977789777678456, −3.37676210779931460729853737988, −2.26083569237403566218453329212, 0,
2.26083569237403566218453329212, 3.37676210779931460729853737988, 4.38188688767321977789777678456, 5.47783897817210982198422627307, 5.81303700718202349463408303074, 7.17903420722688375819191659125, 7.79737690319266234323973828432, 9.438673877808763745213372322799, 10.69417132505853747673508336393
