| L(s) = 1 | + 5.29·3-s + 8i·7-s + 1.00·9-s − 15.8i·11-s − 52.9·13-s − 14i·17-s + 37.0i·19-s + 42.3i·21-s − 152i·23-s − 137.·27-s − 158. i·29-s − 224·31-s − 84.0i·33-s + 243.·37-s − 280·39-s + ⋯ |
| L(s) = 1 | + 1.01·3-s + 0.431i·7-s + 0.0370·9-s − 0.435i·11-s − 1.12·13-s − 0.199i·17-s + 0.447i·19-s + 0.439i·21-s − 1.37i·23-s − 0.980·27-s − 1.01i·29-s − 1.29·31-s − 0.443i·33-s + 1.08·37-s − 1.14·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.734 + 0.678i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9496272166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9496272166\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 5.29T + 27T^{2} \) |
| 7 | \( 1 - 8iT - 343T^{2} \) |
| 11 | \( 1 + 15.8iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 52.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 14iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 37.0iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 152iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 158. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 224T + 2.97e4T^{2} \) |
| 37 | \( 1 - 243.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 70T + 6.89e4T^{2} \) |
| 43 | \( 1 + 439.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 336iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 31.7T + 1.48e5T^{2} \) |
| 59 | \( 1 + 534. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 95.2iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 174.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 72T + 3.57e5T^{2} \) |
| 73 | \( 1 - 294iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 464T + 4.93e5T^{2} \) |
| 83 | \( 1 + 545.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 266T + 7.04e5T^{2} \) |
| 97 | \( 1 - 994iT - 9.12e5T^{2} \) |
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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
−9.472019794690649143179166188418, −8.622005786734525383311999479649, −8.054766369998452507362854110882, −7.12447500476795847599562724242, −6.02206083705438186429303512542, −5.03162542159618158424281366705, −3.85581502070439342016739702332, −2.80091466844116728181909144656, −2.06453418984421874801960520088, −0.19749737365243462584561029094,
1.60425112138716005833359310742, 2.71388434960028552481631792950, 3.61984291338591926458651502044, 4.71083258258858866598364372250, 5.73083116678656916035407862032, 7.16607552221921990541346642477, 7.51692225948040233354951466079, 8.560408759262615670784819746433, 9.356531514583465689660062336812, 9.936511745836110773478552280502
