| L(s) = 1 | + (−3 + 5.19i)2-s + (13.5 + 23.3i)3-s + (46 + 79.6i)4-s + (−195 + 337. i)5-s − 162·6-s − 1.32e3·8-s + (−364.5 + 631. i)9-s + (−1.17e3 − 2.02e3i)10-s + (474 + 820. i)11-s + (−1.24e3 + 2.15e3i)12-s − 5.09e3·13-s − 1.05e4·15-s + (−1.92e3 + 3.33e3i)16-s + (−1.41e4 − 2.45e4i)17-s + (−2.18e3 − 3.78e3i)18-s + (4.31e3 − 7.46e3i)19-s + ⋯ |
| L(s) = 1 | + (−0.265 + 0.459i)2-s + (0.288 + 0.499i)3-s + (0.359 + 0.622i)4-s + (−0.697 + 1.20i)5-s − 0.306·6-s − 0.911·8-s + (−0.166 + 0.288i)9-s + (−0.369 − 0.640i)10-s + (0.107 + 0.185i)11-s + (−0.207 + 0.359i)12-s − 0.643·13-s − 0.805·15-s + (−0.117 + 0.203i)16-s + (−0.700 − 1.21i)17-s + (−0.0883 − 0.153i)18-s + (0.144 − 0.249i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.3809454764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3809454764\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-13.5 - 23.3i)T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + (3 - 5.19i)T + (-64 - 110. i)T^{2} \) |
| 5 | \( 1 + (195 - 337. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 11 | \( 1 + (-474 - 820. i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + 5.09e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + (1.41e4 + 2.45e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-4.31e3 + 7.46e3i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-7.64e3 + 1.32e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 - 3.65e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + (-1.38e5 - 2.39e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (1.34e5 - 2.32e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + 6.29e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.85e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (2.91e5 - 5.05e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-2.14e5 - 3.70e5i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (6.53e5 + 1.13e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (1.50e5 - 2.60e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-2.53e5 - 4.39e5i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 - 5.56e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + (6.84e5 + 1.18e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-3.45e6 + 5.98e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + 4.37e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-4.26e6 + 7.38e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + 8.82e6T + 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.26262708182461601578473697980, −11.49090297472713008032882903574, −10.57091286991103013991824576636, −9.354061256715192055240419883027, −8.242131237703165223035840591183, −7.21700292159831811133761862814, −6.64049284666746278450781268092, −4.74136791384349241964989008100, −3.31411372190724584274570724137, −2.60460805341178118409675584720,
0.11477136610591801454738830990, 1.13688477795705688288287603548, 2.28500547319412939676208910363, 3.96559401296087479731310445687, 5.35361601663078105275789186907, 6.59185907033658295672714208843, 7.965537589909133597367514808772, 8.806861036497482316865743257078, 9.795185575927282098895146736116, 11.03045238438490658635132153917
