| L(s) = 1 | + (265. − 459. i)5-s + (−280. − 485. i)7-s + (−56.0 − 97.1i)11-s + (1.89e3 − 3.28e3i)13-s − 6.16e3·17-s − 6.79e3·19-s + (4.69e4 − 8.13e4i)23-s + (−1.01e5 − 1.75e5i)25-s + (7.91e4 + 1.37e5i)29-s + (1.08e5 − 1.87e5i)31-s − 2.97e5·35-s − 2.10e5·37-s + (1.73e5 − 3.00e5i)41-s + (−3.67e5 − 6.35e5i)43-s + (8.45e4 + 1.46e5i)47-s + ⋯ |
| L(s) = 1 | + (0.948 − 1.64i)5-s + (−0.309 − 0.535i)7-s + (−0.0127 − 0.0220i)11-s + (0.239 − 0.415i)13-s − 0.304·17-s − 0.227·19-s + (0.805 − 1.39i)23-s + (−1.30 − 2.25i)25-s + (0.603 + 1.04i)29-s + (0.652 − 1.13i)31-s − 1.17·35-s − 0.683·37-s + (0.393 − 0.681i)41-s + (−0.704 − 1.21i)43-s + (0.118 + 0.205i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.983261236\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.983261236\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + (-265. + 459. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (280. + 485. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (56.0 + 97.1i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-1.89e3 + 3.28e3i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + 6.16e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 6.79e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + (-4.69e4 + 8.13e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-7.91e4 - 1.37e5i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + (-1.08e5 + 1.87e5i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + 2.10e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-1.73e5 + 3.00e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (3.67e5 + 6.35e5i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 + (-8.45e4 - 1.46e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + 5.56e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (1.15e6 - 2.00e6i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-1.54e6 - 2.67e6i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (2.27e6 - 3.94e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 - 2.84e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.10e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (-5.70e5 - 9.87e5i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (3.54e6 + 6.14e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + 6.42e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (2.08e6 + 3.61e6i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
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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
−9.962044452239863413081254364226, −8.869367370473023401130459948384, −8.459543907718984795782971208639, −6.98861863240936477524691587343, −5.90920167501686753497016234721, −5.00375068130697005612765137685, −4.11191468668703085831354314460, −2.48247470190492403121938610532, −1.21176907704052986495556372843, −0.42668718329920625259736709240,
1.60887259648675965446067673946, 2.64996665654338907209903994140, 3.44552347944930202535936422778, 5.12203801906386371175131657884, 6.32097572507749092090899946190, 6.66807661430728354996364076044, 7.908347637556110710299215941253, 9.291757301221428784540522393857, 9.855572262432197373607278428421, 10.87467753812972149898301209618
