| L(s) = 1 | + 83.0·5-s − 37.1·7-s + 1.95e3·11-s + 4.30e3i·13-s + 7.85e3i·17-s − 6.11e3i·19-s + 1.55e4i·23-s − 8.72e3·25-s + 1.66e3·29-s − 2.78e4·31-s − 3.08e3·35-s − 5.23e4i·37-s − 5.18e4i·41-s + 1.35e5i·43-s − 9.36e4i·47-s + ⋯ |
| L(s) = 1 | + 0.664·5-s − 0.108·7-s + 1.47·11-s + 1.96i·13-s + 1.59i·17-s − 0.891i·19-s + 1.27i·23-s − 0.558·25-s + 0.0684·29-s − 0.935·31-s − 0.0719·35-s − 1.03i·37-s − 0.751i·41-s + 1.70i·43-s − 0.901i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.639 - 0.769i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.639 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.856167954\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.856167954\) |
| \(L(4)\) |
|
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 | \( 1 - 83.0T + 1.56e4T^{2} \) |
| 7 | \( 1 + 37.1T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.95e3T + 1.77e6T^{2} \) |
| 13 | \( 1 - 4.30e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 7.85e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 6.11e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 1.55e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 1.66e3T + 5.94e8T^{2} \) |
| 31 | \( 1 + 2.78e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 5.23e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 5.18e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.35e5iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 9.36e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.04e5T + 2.21e10T^{2} \) |
| 59 | \( 1 - 1.76e5T + 4.21e10T^{2} \) |
| 61 | \( 1 + 926. iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 1.63e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 6.10e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 3.33e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 6.05e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 4.86e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 8.26e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.11e6T + 8.32e11T^{2} \) |
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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
−9.774025980939144307127306196009, −9.320000879659543524324180159765, −8.576647616652244630676011239561, −7.17535056258639198163692368078, −6.48746411500471167860224752983, −5.71153728837941672546940042372, −4.32585532210827236157915289671, −3.65784341775187570684840749591, −1.95995492899681940482432535324, −1.45613958907162594506828949585,
0.35190182563490809927918135556, 1.38205384078196841285742489984, 2.67482667910870866548873502574, 3.63783691737962811202168418736, 4.92941132922145877713437317411, 5.83059606121633976918086752220, 6.63399118867347128528145833955, 7.68804674194439342307669899552, 8.630023587039181370749407485872, 9.595834978035450072658904656600
