| L(s) = 1 | − 12.3·2-s − 27·3-s + 24.9·4-s − 477.·5-s + 333.·6-s + 1.34e3·7-s + 1.27e3·8-s + 729·9-s + 5.91e3·10-s + 6.22e3·11-s − 674.·12-s + 1.09e4·13-s − 1.66e4·14-s + 1.29e4·15-s − 1.89e4·16-s − 3.18e4·17-s − 9.01e3·18-s − 3.64e4·19-s − 1.19e4·20-s − 3.63e4·21-s − 7.69e4·22-s + 3.83e4·23-s − 3.44e4·24-s + 1.50e5·25-s − 1.35e5·26-s − 1.96e4·27-s + 3.35e4·28-s + ⋯ |
| L(s) = 1 | − 1.09·2-s − 0.577·3-s + 0.195·4-s − 1.71·5-s + 0.631·6-s + 1.48·7-s + 0.879·8-s + 0.333·9-s + 1.86·10-s + 1.40·11-s − 0.112·12-s + 1.37·13-s − 1.61·14-s + 0.987·15-s − 1.15·16-s − 1.57·17-s − 0.364·18-s − 1.22·19-s − 0.333·20-s − 0.855·21-s − 1.54·22-s + 0.656·23-s − 0.508·24-s + 1.92·25-s − 1.50·26-s − 0.192·27-s + 0.288·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.7145607435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7145607435\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 59 | \( 1 + 2.05e5T \) |
| good | 2 | \( 1 + 12.3T + 128T^{2} \) |
| 5 | \( 1 + 477.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.34e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 6.22e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.09e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.18e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.64e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.83e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.38e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.56e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.21e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.16e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.48e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 7.48e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.94e6T + 1.17e12T^{2} \) |
| 61 | \( 1 - 1.76e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.25e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.19e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.35e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.91e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.97e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 7.72e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 9.26e6T + 8.07e13T^{2} \) |
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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
−11.12903279020235704545258744620, −10.86013961046493981952669651106, −8.913629382054586826094689804480, −8.516322489286252739619588473442, −7.56408757691146947629541238173, −6.51461793318967654554976716003, −4.45202705040592559313436993850, −4.17077177032802533141134824435, −1.55866135331196834498945166944, −0.61977552223271549743630254131,
0.61977552223271549743630254131, 1.55866135331196834498945166944, 4.17077177032802533141134824435, 4.45202705040592559313436993850, 6.51461793318967654554976716003, 7.56408757691146947629541238173, 8.516322489286252739619588473442, 8.913629382054586826094689804480, 10.86013961046493981952669651106, 11.12903279020235704545258744620
