| L(s) = 1 | − 6.29·2-s − 9·3-s + 7.57·4-s + 82.0·5-s + 56.6·6-s + 51.5·7-s + 153.·8-s + 81·9-s − 516.·10-s + 677.·11-s − 68.1·12-s + 290.·13-s − 324.·14-s − 738.·15-s − 1.20e3·16-s + 1.79e3·17-s − 509.·18-s − 766.·19-s + 621.·20-s − 463.·21-s − 4.26e3·22-s − 3.88e3·23-s − 1.38e3·24-s + 3.61e3·25-s − 1.82e3·26-s − 729·27-s + 390.·28-s + ⋯ |
| L(s) = 1 | − 1.11·2-s − 0.577·3-s + 0.236·4-s + 1.46·5-s + 0.642·6-s + 0.397·7-s + 0.848·8-s + 0.333·9-s − 1.63·10-s + 1.68·11-s − 0.136·12-s + 0.477·13-s − 0.442·14-s − 0.847·15-s − 1.18·16-s + 1.50·17-s − 0.370·18-s − 0.487·19-s + 0.347·20-s − 0.229·21-s − 1.87·22-s − 1.52·23-s − 0.490·24-s + 1.15·25-s − 0.530·26-s − 0.192·27-s + 0.0941·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.384053701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.384053701\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 59 | \( 1 - 3.48e3T \) |
| good | 2 | \( 1 + 6.29T + 32T^{2} \) |
| 5 | \( 1 - 82.0T + 3.12e3T^{2} \) |
| 7 | \( 1 - 51.5T + 1.68e4T^{2} \) |
| 11 | \( 1 - 677.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 290.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.79e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 766.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.88e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.72e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.41e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.02e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.30e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.17e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 3.79e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 5.74e3T + 4.18e8T^{2} \) |
| 61 | \( 1 + 3.45e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.86e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.01e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.28e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.30e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.67e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.03e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.50e4T + 8.58e9T^{2} \) |
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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
−11.55800941665129242693644783631, −10.44895498239637623489998963161, −9.748385965279275943176905707441, −9.049170209256535329814329083858, −7.86845626215502830910492757417, −6.48944091338860879891763753306, −5.67235855611221371363276848053, −4.16494482603061478615931052021, −1.80591707110677906970293135019, −1.01810232019338471843676685262,
1.01810232019338471843676685262, 1.80591707110677906970293135019, 4.16494482603061478615931052021, 5.67235855611221371363276848053, 6.48944091338860879891763753306, 7.86845626215502830910492757417, 9.049170209256535329814329083858, 9.748385965279275943176905707441, 10.44895498239637623489998963161, 11.55800941665129242693644783631
