| L(s) = 1 | + 124. i·5-s − 646.·7-s + 5.13e3i·11-s + 1.34e4i·13-s + 2.13e4·17-s − 1.22e4i·19-s − 1.51e4·23-s + 6.26e4·25-s − 1.32e5i·29-s − 8.10e4·31-s − 8.02e4i·35-s + 4.29e5i·37-s − 3.14e5·41-s + 6.86e5i·43-s + 3.51e5·47-s + ⋯ |
| L(s) = 1 | + 0.444i·5-s − 0.712·7-s + 1.16i·11-s + 1.70i·13-s + 1.05·17-s − 0.411i·19-s − 0.259·23-s + 0.802·25-s − 1.01i·29-s − 0.488·31-s − 0.316i·35-s + 1.39i·37-s − 0.713·41-s + 1.31i·43-s + 0.493·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0611i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0611i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.8522781729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8522781729\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 124. iT - 7.81e4T^{2} \) |
| 7 | \( 1 + 646.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.13e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 - 1.34e4iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 2.13e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.22e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 1.51e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.32e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 8.10e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.29e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 3.14e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.86e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 3.51e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.02e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 2.99e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 3.72e5iT - 3.14e12T^{2} \) |
| 67 | \( 1 - 1.57e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 3.21e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.96e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.94e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 9.26e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 1.28e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.36e7T + 8.07e13T^{2} \) |
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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
−11.11881077030284248858716145031, −9.813703377911628087752386958606, −9.550550198109934859266425929199, −8.147428559056300675261514605490, −6.94997213570448065996901094643, −6.48019373314178977090589679227, −4.97883351086760644844623622875, −3.91769048488019919840699990297, −2.69077729547533761778489067872, −1.50481869219109587049769919834,
0.20659826181636253448635916726, 1.12264089282877568293602065462, 2.93070938895119424872269400758, 3.65645318634577664086304277848, 5.35175760917878044969551995353, 5.87038029996860810323467148556, 7.26851905309049228334732288182, 8.264658737450114303304813374219, 9.065201340829290284621964308529, 10.24087222770989490475941963504
