| L(s) = 1 | + (−3.80 + 1.23i)2-s + (10.0 + 13.8i)3-s + (12.9 − 9.40i)4-s + (−52.7 + 18.5i)5-s + (−55.2 − 40.1i)6-s + 107. i·7-s + (−37.6 + 51.7i)8-s + (−15.1 + 46.4i)9-s + (177. − 135. i)10-s + (−33.7 − 103. i)11-s + (259. + 84.4i)12-s + (−458. − 148. i)13-s + (−133. − 409. i)14-s + (−785. − 542. i)15-s + (79.1 − 243. i)16-s + (−1.08e3 + 1.49e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 + 0.218i)2-s + (0.644 + 0.886i)3-s + (0.404 − 0.293i)4-s + (−0.943 + 0.331i)5-s + (−0.626 − 0.455i)6-s + 0.830i·7-s + (−0.207 + 0.286i)8-s + (−0.0621 + 0.191i)9-s + (0.561 − 0.429i)10-s + (−0.0841 − 0.258i)11-s + (0.521 + 0.169i)12-s + (−0.752 − 0.244i)13-s + (−0.181 − 0.558i)14-s + (−0.901 − 0.622i)15-s + (0.0772 − 0.237i)16-s + (−0.911 + 1.25i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.110i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0344386 - 0.621295i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0344386 - 0.621295i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.80 - 1.23i)T \) |
| 5 | \( 1 + (52.7 - 18.5i)T \) |
| good | 3 | \( 1 + (-10.0 - 13.8i)T + (-75.0 + 231. i)T^{2} \) |
| 7 | \( 1 - 107. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (33.7 + 103. i)T + (-1.30e5 + 9.46e4i)T^{2} \) |
| 13 | \( 1 + (458. + 148. i)T + (3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (1.08e3 - 1.49e3i)T + (-4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (2.18e3 + 1.58e3i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + (840. - 272. i)T + (5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (3.02e3 - 2.19e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-5.00e3 - 3.63e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-5.79e3 - 1.88e3i)T + (5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (5.68e3 - 1.74e4i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 - 9.07e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-2.98e3 - 4.10e3i)T + (-7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-4.69e3 - 6.45e3i)T + (-1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-3.42e3 + 1.05e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-1.07e3 - 3.31e3i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-1.66e4 + 2.29e4i)T + (-4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (5.16e4 - 3.75e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-1.81e4 + 5.90e3i)T + (1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (7.36e4 - 5.35e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-732. + 1.00e3i)T + (-1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (1.47e4 + 4.53e4i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (3.18e4 + 4.38e4i)T + (-2.65e9 + 8.16e9i)T^{2} \) |
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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
−15.16139367767551792197315025679, −14.75826670959424464276363455196, −12.71261298179347286305345106402, −11.33111817787631160717783367516, −10.26029516922962129225671254786, −8.934910802411794505871648871912, −8.208278500809418694758499126610, −6.52911655323559500870621992579, −4.43091026360466276362731294545, −2.75472578546077917014695142473,
0.34784041315158435543638599760, 2.21444261004965921749854727841, 4.23054309251485724713196208030, 7.01351310922364204346635337796, 7.71945455665927313317241992169, 8.796618076175769409697692304810, 10.33535587314328936069728233391, 11.69077999153872038622693856561, 12.72031884856023470960862378612, 13.79884802723831659310512287696
