| L(s) = 1 | + 227. i·5-s + 1.65e3·13-s − 7.61e3i·17-s − 3.62e4·25-s + 4.86e4i·29-s − 5.55e4·37-s + 1.36e5i·41-s − 1.17e5·49-s − 1.88e5i·53-s − 2.34e5·61-s + 3.77e5i·65-s − 6.50e5·73-s + 1.73e6·85-s − 1.18e6i·89-s − 1.07e6·97-s + ⋯ |
| L(s) = 1 | + 1.82i·5-s + 0.753·13-s − 1.54i·17-s − 2.31·25-s + 1.99i·29-s − 1.09·37-s + 1.98i·41-s − 49-s − 1.26i·53-s − 1.03·61-s + 1.37i·65-s − 1.67·73-s + 2.82·85-s − 1.67i·89-s − 1.18·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.2044508826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2044508826\) |
| \(L(4)\) |
|
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 | \( 1 - 227. iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.77e6T^{2} \) |
| 13 | \( 1 - 1.65e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 7.61e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 4.70e7T^{2} \) |
| 23 | \( 1 - 1.48e8T^{2} \) |
| 29 | \( 1 - 4.86e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 8.87e8T^{2} \) |
| 37 | \( 1 + 5.55e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 1.36e5iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.88e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 4.21e10T^{2} \) |
| 61 | \( 1 + 2.34e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 9.04e10T^{2} \) |
| 71 | \( 1 - 1.28e11T^{2} \) |
| 73 | \( 1 + 6.50e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 2.43e11T^{2} \) |
| 83 | \( 1 - 3.26e11T^{2} \) |
| 89 | \( 1 + 1.18e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.07e6T + 8.32e11T^{2} \) |
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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
−10.39934699740606478570539604670, −9.649497768930723585307979169984, −8.563934026277404836250283046165, −7.38945700717419745174281332794, −6.86890164649612799650498225825, −6.02749082172794430336958274210, −4.82529556342047172655309668361, −3.36027773590047097474228740676, −2.93862663752400821929064888493, −1.59916567366805774421020719954,
0.04159620247307361740760931604, 1.11935845168198370358363770278, 1.96599528653488091966552358843, 3.76108810636034909155153953384, 4.43665790035729161973690246165, 5.53405480552686262572126082769, 6.19049394947227167948484254954, 7.70114198783562487123628078012, 8.468210837974714147963467422978, 8.979142710314749034383651656759
