| L(s) = 1 | + (−11.9 + 45.2i)3-s − 304.·5-s − 76.4i·7-s + (−1.90e3 − 1.08e3i)9-s − 2.83e3i·11-s − 3.46e3i·13-s + (3.64e3 − 1.37e4i)15-s + 2.15e3i·17-s + 5.30e4·19-s + (3.45e3 + 915. i)21-s − 6.37e4·23-s + 1.44e4·25-s + (7.16e4 − 7.29e4i)27-s + 9.17e4·29-s − 1.08e5i·31-s + ⋯ |
| L(s) = 1 | + (−0.256 + 0.966i)3-s − 1.08·5-s − 0.0842i·7-s + (−0.868 − 0.494i)9-s − 0.642i·11-s − 0.437i·13-s + (0.278 − 1.05i)15-s + 0.106i·17-s + 1.77·19-s + (0.0814 + 0.0215i)21-s − 1.09·23-s + 0.185·25-s + (0.700 − 0.713i)27-s + 0.698·29-s − 0.655i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00290 - 0.999i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.00290 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.002524669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.002524669\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (11.9 - 45.2i)T \) |
| good | 5 | \( 1 + 304.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 76.4iT - 8.23e5T^{2} \) |
| 11 | \( 1 + 2.83e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 + 3.46e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 2.15e3iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 5.30e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.37e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 9.17e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.08e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 - 4.59e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 2.01e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 - 3.11e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.00e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.56e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.91e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 2.39e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 4.45e4T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.04e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.55e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.36e4iT - 1.92e13T^{2} \) |
| 83 | \( 1 - 5.95e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 1.21e7iT - 4.42e13T^{2} \) |
| 97 | \( 1 - 1.39e7T + 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.64559370897936204091332855329, −10.51720556782553119988477358724, −9.656093154666036918906926310598, −8.449957319823299864547695839276, −7.64358173202903917452888043979, −6.11983045648302325979811549246, −5.01576090847060456675236592456, −3.87982482375461771980846610166, −3.06315521432468539958090956393, −0.75412529666616781442468761199,
0.39717112301032052650590164756, 1.72559967398684334792807227750, 3.18718958199100570063363186947, 4.57720204640098097894247392386, 5.86225180569509372807489348997, 7.17271614959533583061702738815, 7.66783656528196717214977343596, 8.762798035734200164171093854910, 10.08154141170456552929584500661, 11.48364407859586577497526243751
