| L(s) = 1 | + (−15.6 + 3.46i)2-s + (−118. − 38.4i)3-s + (231. − 108. i)4-s + (−243. − 575. i)5-s + (1.98e3 + 190. i)6-s − 2.34e3i·7-s + (−3.24e3 + 2.49e3i)8-s + (7.21e3 + 5.24e3i)9-s + (5.80e3 + 8.14e3i)10-s + (2.70e3 + 3.72e3i)11-s + (−3.16e4 + 3.89e3i)12-s + (1.46e3 + 1.06e3i)13-s + (8.13e3 + 3.66e4i)14-s + (6.71e3 + 7.74e4i)15-s + (4.20e4 − 5.02e4i)16-s + (−5.66e3 − 1.74e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.976 + 0.216i)2-s + (−1.46 − 0.474i)3-s + (0.906 − 0.422i)4-s + (−0.390 − 0.920i)5-s + (1.52 + 0.146i)6-s − 0.977i·7-s + (−0.793 + 0.609i)8-s + (1.09 + 0.799i)9-s + (0.580 + 0.814i)10-s + (0.184 + 0.254i)11-s + (−1.52 + 0.187i)12-s + (0.0513 + 0.0372i)13-s + (0.211 + 0.953i)14-s + (0.132 + 1.53i)15-s + (0.642 − 0.766i)16-s + (−0.0678 − 0.208i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.261806 + 0.0140161i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.261806 + 0.0140161i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (15.6 - 3.46i)T \) |
| 5 | \( 1 + (243. + 575. i)T \) |
| good | 3 | \( 1 + (118. + 38.4i)T + (5.30e3 + 3.85e3i)T^{2} \) |
| 7 | \( 1 + 2.34e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + (-2.70e3 - 3.72e3i)T + (-6.62e7 + 2.03e8i)T^{2} \) |
| 13 | \( 1 + (-1.46e3 - 1.06e3i)T + (2.52e8 + 7.75e8i)T^{2} \) |
| 17 | \( 1 + (5.66e3 + 1.74e4i)T + (-5.64e9 + 4.10e9i)T^{2} \) |
| 19 | \( 1 + (1.39e5 - 4.54e4i)T + (1.37e10 - 9.98e9i)T^{2} \) |
| 23 | \( 1 + (2.35e5 + 3.24e5i)T + (-2.41e10 + 7.44e10i)T^{2} \) |
| 29 | \( 1 + (2.03e5 - 6.26e5i)T + (-4.04e11 - 2.94e11i)T^{2} \) |
| 31 | \( 1 + (1.22e6 - 3.99e5i)T + (6.90e11 - 5.01e11i)T^{2} \) |
| 37 | \( 1 + (9.23e4 + 6.70e4i)T + (1.08e12 + 3.34e12i)T^{2} \) |
| 41 | \( 1 + (-4.04e6 - 2.93e6i)T + (2.46e12 + 7.59e12i)T^{2} \) |
| 43 | \( 1 + 4.03e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (8.78e5 + 2.85e5i)T + (1.92e13 + 1.39e13i)T^{2} \) |
| 53 | \( 1 + (4.56e6 - 1.40e7i)T + (-5.03e13 - 3.65e13i)T^{2} \) |
| 59 | \( 1 + (5.55e6 - 7.64e6i)T + (-4.53e13 - 1.39e14i)T^{2} \) |
| 61 | \( 1 + (-9.73e6 + 7.07e6i)T + (5.92e13 - 1.82e14i)T^{2} \) |
| 67 | \( 1 + (-9.32e6 + 3.02e6i)T + (3.28e14 - 2.38e14i)T^{2} \) |
| 71 | \( 1 + (-9.65e5 - 3.13e5i)T + (5.22e14 + 3.79e14i)T^{2} \) |
| 73 | \( 1 + (3.45e7 - 2.51e7i)T + (2.49e14 - 7.66e14i)T^{2} \) |
| 79 | \( 1 + (1.65e7 + 5.37e6i)T + (1.22e15 + 8.91e14i)T^{2} \) |
| 83 | \( 1 + (-5.96e7 + 1.93e7i)T + (1.82e15 - 1.32e15i)T^{2} \) |
| 89 | \( 1 + (-1.47e6 + 1.07e6i)T + (1.21e15 - 3.74e15i)T^{2} \) |
| 97 | \( 1 + (-2.32e7 + 7.14e7i)T + (-6.34e15 - 4.60e15i)T^{2} \) |
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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
−12.15635350923642305569430844335, −11.05713176096000345362928356422, −10.39504950732808363568001173486, −8.965004829252493918858202866898, −7.69561202966009860976831988433, −6.77541098467615350183057051334, −5.68842384689760336320265838553, −4.34718542254681719284527868759, −1.60040946643246183036359929867, −0.58642059564465278314693709176,
0.23572855343680171570446562473, 2.17662215530756903371160755840, 3.83216967136126851869064155712, 5.74673677219109022532913625669, 6.47027737174500939517975212005, 7.82037208005718666115355250765, 9.255296958211823184882178814587, 10.31936921852630400625265669612, 11.24920011633562240824681688085, 11.62645914777986661080214665301
