| L(s) = 1 | + (−1.23 + 3.80i)2-s + (−12.9 − 9.40i)4-s + (105. − 34.2i)5-s + (−53.1 + 73.1i)7-s + (51.7 − 37.6i)8-s + 443. i·10-s + (272. + 294. i)11-s + (−429. − 139. i)13-s + (−212. − 292. i)14-s + (79.1 + 243. i)16-s + (−26.5 − 81.7i)17-s + (1.12e3 + 1.54e3i)19-s + (−1.68e3 − 547. i)20-s + (−1.45e3 + 672. i)22-s + 473. i·23-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + (−0.404 − 0.293i)4-s + (1.88 − 0.612i)5-s + (−0.409 + 0.563i)7-s + (0.286 − 0.207i)8-s + 1.40i·10-s + (0.679 + 0.734i)11-s + (−0.704 − 0.228i)13-s + (−0.289 − 0.398i)14-s + (0.0772 + 0.237i)16-s + (−0.0222 − 0.0686i)17-s + (0.713 + 0.982i)19-s + (−0.942 − 0.306i)20-s + (−0.642 + 0.296i)22-s + 0.186i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 - 0.868i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.496 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.397439601\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.397439601\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 - 3.80i)T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (-272. - 294. i)T \) |
| good | 5 | \( 1 + (-105. + 34.2i)T + (2.52e3 - 1.83e3i)T^{2} \) |
| 7 | \( 1 + (53.1 - 73.1i)T + (-5.19e3 - 1.59e4i)T^{2} \) |
| 13 | \( 1 + (429. + 139. i)T + (3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (26.5 + 81.7i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-1.12e3 - 1.54e3i)T + (-7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 - 473. iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (158. + 114. i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-2.40e3 + 7.38e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-7.48e3 - 5.44e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-4.49e3 + 3.26e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 - 8.15e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-4.09e3 - 5.64e3i)T + (-7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-1.93e4 - 6.27e3i)T + (3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (1.88e4 - 2.59e4i)T + (-2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (3.85e4 - 1.25e4i)T + (6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + 494.T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-1.93e3 + 629. i)T + (1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-2.04e4 + 2.80e4i)T + (-6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-6.03e4 - 1.95e4i)T + (2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-2.80e4 - 8.63e4i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + 1.25e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (-4.54e4 + 1.39e5i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−12.05359945257886248878770979419, −10.22939847036175942368295707983, −9.559034697417508381323598852629, −9.099258099386398591504081779423, −7.66518416209833737255456954187, −6.28867103815697188909810986791, −5.73499026363399942567437779167, −4.62656982440925232527573454406, −2.45713095295174969429145623175, −1.21072948765508678973481896355,
0.932928184857558584583372518572, 2.25820613027934307422678783198, 3.30855886352936953862017000007, 5.04039113466533695580060224946, 6.27386207539794389667586873194, 7.13672844477804090815088463801, 8.943265800073847529573862715322, 9.586009486619894087715116237194, 10.38212316496473418969666315803, 11.16079945214766979500442167145
