| 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
−13.79884802723831659310512287696, −12.72031884856023470960862378612, −11.69077999153872038622693856561, −10.33535587314328936069728233391, −8.796618076175769409697692304810, −7.71945455665927313317241992169, −7.01351310922364204346635337796, −4.23054309251485724713196208030, −2.21444261004965921749854727841, −0.34784041315158435543638599760,
2.75472578546077917014695142473, 4.43091026360466276362731294545, 6.52911655323559500870621992579, 8.208278500809418694758499126610, 8.934910802411794505871648871912, 10.26029516922962129225671254786, 11.33111817787631160717783367516, 12.71261298179347286305345106402, 14.75826670959424464276363455196, 15.16139367767551792197315025679
