| L(s) = 1 | + (−6.90 − 5.77i)3-s + (11.0 + 13.1i)5-s + (7.77 − 2.83i)7-s + (14.3 + 79.7i)9-s + (44.3 − 52.8i)11-s + (−4.19 − 23.8i)13-s + (−0.283 − 154. i)15-s + (213. − 123. i)17-s + (175. − 304. i)19-s + (−70.0 − 25.3i)21-s + (275. − 756. i)23-s + (57.1 − 323. i)25-s + (361. − 633. i)27-s + (−348. − 61.3i)29-s + (634. + 231. i)31-s + ⋯ |
| L(s) = 1 | + (−0.767 − 0.641i)3-s + (0.442 + 0.527i)5-s + (0.158 − 0.0577i)7-s + (0.177 + 0.984i)9-s + (0.366 − 0.436i)11-s + (−0.0248 − 0.140i)13-s + (−0.00125 − 0.688i)15-s + (0.738 − 0.426i)17-s + (0.486 − 0.842i)19-s + (−0.158 − 0.0574i)21-s + (0.520 − 1.43i)23-s + (0.0914 − 0.518i)25-s + (0.495 − 0.868i)27-s + (−0.413 − 0.0729i)29-s + (0.660 + 0.240i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.601 + 0.798i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.601 + 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.31365 - 0.655234i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31365 - 0.655234i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (6.90 + 5.77i)T \) |
| good | 5 | \( 1 + (-11.0 - 13.1i)T + (-108. + 615. i)T^{2} \) |
| 7 | \( 1 + (-7.77 + 2.83i)T + (1.83e3 - 1.54e3i)T^{2} \) |
| 11 | \( 1 + (-44.3 + 52.8i)T + (-2.54e3 - 1.44e4i)T^{2} \) |
| 13 | \( 1 + (4.19 + 23.8i)T + (-2.68e4 + 9.76e3i)T^{2} \) |
| 17 | \( 1 + (-213. + 123. i)T + (4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-175. + 304. i)T + (-6.51e4 - 1.12e5i)T^{2} \) |
| 23 | \( 1 + (-275. + 756. i)T + (-2.14e5 - 1.79e5i)T^{2} \) |
| 29 | \( 1 + (348. + 61.3i)T + (6.64e5 + 2.41e5i)T^{2} \) |
| 31 | \( 1 + (-634. - 231. i)T + (7.07e5 + 5.93e5i)T^{2} \) |
| 37 | \( 1 + (-870. - 1.50e3i)T + (-9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 + (-629. + 111. i)T + (2.65e6 - 9.66e5i)T^{2} \) |
| 43 | \( 1 + (-1.14e3 - 964. i)T + (5.93e5 + 3.36e6i)T^{2} \) |
| 47 | \( 1 + (649. + 1.78e3i)T + (-3.73e6 + 3.13e6i)T^{2} \) |
| 53 | \( 1 + 1.72e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (442. + 527. i)T + (-2.10e6 + 1.19e7i)T^{2} \) |
| 61 | \( 1 + (-995. + 362. i)T + (1.06e7 - 8.89e6i)T^{2} \) |
| 67 | \( 1 + (-1.01e3 - 5.78e3i)T + (-1.89e7 + 6.89e6i)T^{2} \) |
| 71 | \( 1 + (-3.88e3 + 2.24e3i)T + (1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 + (725. - 1.25e3i)T + (-1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (173. - 986. i)T + (-3.66e7 - 1.33e7i)T^{2} \) |
| 83 | \( 1 + (1.71e3 + 301. i)T + (4.45e7 + 1.62e7i)T^{2} \) |
| 89 | \( 1 + (1.13e3 + 653. i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 + (-1.04e4 - 8.80e3i)T + (1.53e7 + 8.71e7i)T^{2} \) |
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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
−12.80521030586913285013275431072, −11.72221911151138506856043583068, −10.88693945773114855472496719128, −9.835507242071804236035930081131, −8.284080936146611192852777870121, −7.00611519985519169214792402931, −6.13171973762778501329862799374, −4.84067503440596614175877951976, −2.69734768484498929205778042654, −0.867508424585515151380461863210,
1.31408475367742194455233420344, 3.74127222774714672460955802095, 5.12438315153267325006678283022, 6.00768537427240605888789357344, 7.55764317794110178391192820767, 9.199638553781413391133288647834, 9.838604820836200366869772600192, 11.07119008295878966596859714794, 12.03779967532828778342325947833, 12.93325023512546340591552076630
