| L(s) = 1 | + (−1.17 − 1.61i)2-s + (4.16 + 1.35i)3-s + (−1.23 + 3.80i)4-s + (−2.70 − 8.33i)6-s − 31.2i·7-s + (7.60 − 2.47i)8-s + (−6.32 − 4.59i)9-s + (12.0 − 8.76i)11-s + (−10.2 + 14.1i)12-s + (−25.0 + 34.5i)13-s + (−50.4 + 36.6i)14-s + (−12.9 − 9.40i)16-s + (−27.6 + 8.99i)17-s + 15.6i·18-s + (15.9 + 48.9i)19-s + ⋯ |
| L(s) = 1 | + (−0.415 − 0.572i)2-s + (0.801 + 0.260i)3-s + (−0.154 + 0.475i)4-s + (−0.184 − 0.566i)6-s − 1.68i·7-s + (0.336 − 0.109i)8-s + (−0.234 − 0.170i)9-s + (0.330 − 0.240i)11-s + (−0.247 + 0.340i)12-s + (−0.535 + 0.736i)13-s + (−0.963 + 0.700i)14-s + (−0.202 − 0.146i)16-s + (−0.394 + 0.128i)17-s + 0.204i·18-s + (0.192 + 0.591i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.795 + 0.605i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.795 + 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.390472 - 1.15807i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.390472 - 1.15807i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.17 + 1.61i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (-4.16 - 1.35i)T + (21.8 + 15.8i)T^{2} \) |
| 7 | \( 1 + 31.2iT - 343T^{2} \) |
| 11 | \( 1 + (-12.0 + 8.76i)T + (411. - 1.26e3i)T^{2} \) |
| 13 | \( 1 + (25.0 - 34.5i)T + (-678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (27.6 - 8.99i)T + (3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-15.9 - 48.9i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + (109. + 150. i)T + (-3.75e3 + 1.15e4i)T^{2} \) |
| 29 | \( 1 + (-23.7 + 73.1i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (80.3 + 247. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (70.7 - 97.3i)T + (-1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-248. - 180. i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 + 504. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-186. - 60.7i)T + (8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (430. + 139. i)T + (1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-51.9 - 37.7i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-110. + 80.3i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + (-223. + 72.6i)T + (2.43e5 - 1.76e5i)T^{2} \) |
| 71 | \( 1 + (-166. + 512. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-614. - 846. i)T + (-1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (109. - 335. i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (928. - 301. i)T + (4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 + (-891. + 647. i)T + (2.17e5 - 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-1.37e3 - 448. i)T + (7.38e5 + 5.36e5i)T^{2} \) |
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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
−11.08574942925438556376181129157, −10.14008298666840492234041940915, −9.471224550414346154076867200971, −8.393122890178797783477925138653, −7.55918150969435008520432348340, −6.40337548979765730745048844898, −4.30963985587024155164040959223, −3.71711399065658104894999846482, −2.20715127042021144039548729935, −0.48149065012809631604893314968,
1.94701347943575281520372598517, 3.05873706342599196463783995640, 5.08386812891508348143637156290, 5.91565366758390276572045777802, 7.28072413641202435572448333573, 8.178353448772135418670394698287, 8.999561427636184426626795888075, 9.566345083800639437396657679415, 11.01437055725028294545016801017, 12.08316028199247190425708664110
