| L(s) = 1 | − 81i·3-s − 573. i·5-s + 2.18e3·7-s − 6.56e3·9-s + 2.51e4i·11-s + 1.70e5i·13-s − 4.64e4·15-s + 4.61e5·17-s − 1.65e5i·19-s − 1.77e5i·21-s + 9.53e5·23-s + 1.62e6·25-s + 5.31e5i·27-s − 1.12e6i·29-s − 8.86e5·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.410i·5-s + 0.343·7-s − 0.333·9-s + 0.518i·11-s + 1.65i·13-s − 0.236·15-s + 1.33·17-s − 0.291i·19-s − 0.198i·21-s + 0.710·23-s + 0.831·25-s + 0.192i·27-s − 0.295i·29-s − 0.172·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.416143736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.416143736\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 81iT \) |
| good | 5 | \( 1 + 573. iT - 1.95e6T^{2} \) |
| 7 | \( 1 - 2.18e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.51e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 - 1.70e5iT - 1.06e10T^{2} \) |
| 17 | \( 1 - 4.61e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 1.65e5iT - 3.22e11T^{2} \) |
| 23 | \( 1 - 9.53e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.12e6iT - 1.45e13T^{2} \) |
| 31 | \( 1 + 8.86e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.04e7iT - 1.29e14T^{2} \) |
| 41 | \( 1 + 2.74e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.48e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 + 5.83e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.24e7iT - 3.29e15T^{2} \) |
| 59 | \( 1 + 5.50e7iT - 8.66e15T^{2} \) |
| 61 | \( 1 - 3.22e7iT - 1.16e16T^{2} \) |
| 67 | \( 1 - 1.97e8iT - 2.72e16T^{2} \) |
| 71 | \( 1 + 1.31e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.88e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 6.76e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 3.73e8iT - 1.86e17T^{2} \) |
| 89 | \( 1 + 2.50e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 7.73e8T + 7.60e17T^{2} \) |
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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
−9.885569559311419381767711029433, −9.091321886452753547111095686158, −8.184042977322548698334012909971, −7.22487757157286017400674139928, −6.47135971580965376913598928458, −5.21879820930896657961855743121, −4.41503148006629932939816861920, −3.08312611655763436050563828216, −1.79619551755466535626043129665, −1.14201622919820316463024833640,
0.26348877431804642025320863907, 1.38860422970214596896886575295, 3.04219100655899436852220365536, 3.41020378060981799694481800323, 4.99601315764445522811109358446, 5.55504452645716111243494489541, 6.76923628290531768654441738507, 7.944484750740884075108753788496, 8.533549111786261978051242953202, 9.836656167104073214682986309046
