| L(s) = 1 | + (5.59 + 0.843i)2-s + (29.6 + 43.5i)3-s + (30.5 + 9.43i)4-s + (191. − 14.3i)5-s + (129. + 268. i)6-s + (−340. + 44.5i)7-s + (163. + 78.5i)8-s + (−746. + 1.90e3i)9-s + (1.08e3 + 81.0i)10-s + (347. + 885. i)11-s + (496. + 1.61e3i)12-s + (−2.55e3 − 2.03e3i)13-s + (−1.93e3 − 37.2i)14-s + (6.29e3 + 7.89e3i)15-s + (846. + 576. i)16-s + (5.27e3 − 5.68e3i)17-s + ⋯ |
| L(s) = 1 | + (0.699 + 0.105i)2-s + (1.09 + 1.61i)3-s + (0.477 + 0.147i)4-s + (1.53 − 0.114i)5-s + (0.598 + 1.24i)6-s + (−0.991 + 0.130i)7-s + (0.318 + 0.153i)8-s + (−1.02 + 2.61i)9-s + (1.08 + 0.0810i)10-s + (0.260 + 0.664i)11-s + (0.287 + 0.931i)12-s + (−1.16 − 0.926i)13-s + (−0.706 − 0.0135i)14-s + (1.86 + 2.34i)15-s + (0.206 + 0.140i)16-s + (1.07 − 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.278 - 0.960i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.278 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.89567 + 3.85425i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.89567 + 3.85425i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-5.59 - 0.843i)T \) |
| 7 | \( 1 + (340. - 44.5i)T \) |
| good | 3 | \( 1 + (-29.6 - 43.5i)T + (-266. + 678. i)T^{2} \) |
| 5 | \( 1 + (-191. + 14.3i)T + (1.54e4 - 2.32e3i)T^{2} \) |
| 11 | \( 1 + (-347. - 885. i)T + (-1.29e6 + 1.20e6i)T^{2} \) |
| 13 | \( 1 + (2.55e3 + 2.03e3i)T + (1.07e6 + 4.70e6i)T^{2} \) |
| 17 | \( 1 + (-5.27e3 + 5.68e3i)T + (-1.80e6 - 2.40e7i)T^{2} \) |
| 19 | \( 1 + (-5.76e3 + 3.32e3i)T + (2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (3.63e3 - 3.37e3i)T + (1.10e7 - 1.47e8i)T^{2} \) |
| 29 | \( 1 + (-799. + 3.50e3i)T + (-5.35e8 - 2.58e8i)T^{2} \) |
| 31 | \( 1 + (-2.20e4 - 1.27e4i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (2.35e4 - 7.24e3i)T + (2.11e9 - 1.44e9i)T^{2} \) |
| 41 | \( 1 + (-2.79e4 + 5.81e4i)T + (-2.96e9 - 3.71e9i)T^{2} \) |
| 43 | \( 1 + (-2.65e4 + 1.27e4i)T + (3.94e9 - 4.94e9i)T^{2} \) |
| 47 | \( 1 + (-750. + 4.97e3i)T + (-1.03e10 - 3.17e9i)T^{2} \) |
| 53 | \( 1 + (1.05e4 + 3.26e3i)T + (1.83e10 + 1.24e10i)T^{2} \) |
| 59 | \( 1 + (2.39e5 + 1.79e4i)T + (4.17e10 + 6.28e9i)T^{2} \) |
| 61 | \( 1 + (-2.27e4 - 7.36e4i)T + (-4.25e10 + 2.90e10i)T^{2} \) |
| 67 | \( 1 + (2.05e5 - 3.55e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + (8.84e4 + 3.87e5i)T + (-1.15e11 + 5.55e10i)T^{2} \) |
| 73 | \( 1 + (3.26e4 + 2.16e5i)T + (-1.44e11 + 4.46e10i)T^{2} \) |
| 79 | \( 1 + (-3.62e5 - 6.27e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-3.99e5 + 3.18e5i)T + (7.27e10 - 3.18e11i)T^{2} \) |
| 89 | \( 1 + (-6.56e5 - 2.57e5i)T + (3.64e11 + 3.38e11i)T^{2} \) |
| 97 | \( 1 + 1.59e4iT - 8.32e11T^{2} \) |
| 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
−13.55725627684111672683359279639, −12.25070129986717203633937955708, −10.32900055900113563777211014841, −9.795014904746119636601202172072, −9.214339243916659262751317917144, −7.44685789116763813918440578277, −5.59937547304702460546717405522, −4.86580312320605687484522042452, −3.19924471730453317690130857946, −2.47158748156827473043784566760,
1.27084736987094130189082426036, 2.33726043071663045870048890484, 3.36033501625032136375023345350, 5.95868630921051462776439392136, 6.48548769382941719666641872309, 7.67858863680133916618404528688, 9.189168094169067431329043271523, 9.995021815852661317990633850290, 12.04223785453580998582278593518, 12.69396666537942224203633315244
