| L(s) = 1 | + (−5.64 − 0.375i)2-s + (−5.80 + 10.0i)3-s + (31.7 + 4.24i)4-s + (−48.4 + 83.8i)5-s + (36.5 − 54.5i)6-s + 77.0i·7-s + (−177. − 35.8i)8-s + (54.1 + 93.7i)9-s + (304. − 455. i)10-s + 612. i·11-s + (−226. + 294. i)12-s + (784. − 453. i)13-s + (28.9 − 434. i)14-s + (−562. − 973. i)15-s + (988. + 269. i)16-s + (−469. + 812. i)17-s + ⋯ |
| L(s) = 1 | + (−0.997 − 0.0664i)2-s + (−0.372 + 0.644i)3-s + (0.991 + 0.132i)4-s + (−0.866 + 1.50i)5-s + (0.414 − 0.618i)6-s + 0.594i·7-s + (−0.980 − 0.198i)8-s + (0.222 + 0.385i)9-s + (0.963 − 1.43i)10-s + 1.52i·11-s + (−0.454 + 0.589i)12-s + (1.28 − 0.743i)13-s + (0.0394 − 0.592i)14-s + (−0.644 − 1.11i)15-s + (0.964 + 0.262i)16-s + (−0.393 + 0.682i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.885 + 0.464i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.885 + 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.144686 - 0.587211i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.144686 - 0.587211i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (5.64 + 0.375i)T \) |
| 19 | \( 1 + (1.57e3 + 8.46i)T \) |
| good | 3 | \( 1 + (5.80 - 10.0i)T + (-121.5 - 210. i)T^{2} \) |
| 5 | \( 1 + (48.4 - 83.8i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 - 77.0iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 612. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (-784. + 453. i)T + (1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 + (469. - 812. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 23 | \( 1 + (156. - 90.1i)T + (3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-5.13e3 + 2.96e3i)T + (1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 + 4.14e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 573. iT - 6.93e7T^{2} \) |
| 41 | \( 1 + (-2.74e3 - 1.58e3i)T + (5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-1.17e4 - 6.80e3i)T + (7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-1.77e3 + 1.02e3i)T + (1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-1.46e4 + 8.44e3i)T + (2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-1.73e4 + 3.01e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-6.67e3 - 1.15e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-907. - 1.57e3i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (3.84e4 - 6.66e4i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (-3.05e4 + 5.28e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (3.40e4 - 5.90e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + 9.86e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + (9.28e4 - 5.35e4i)T + (2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + (4.25e4 + 2.45e4i)T + (4.29e9 + 7.43e9i)T^{2} \) |
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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
−14.82035443171441745195695643741, −12.69640584864745527285257058172, −11.43382038938944948692883721880, −10.68028163537087437734885987117, −10.04978240849756908974999850660, −8.409014269119734262230947370124, −7.31552262156412850165439574698, −6.16377659545988711166984734844, −3.97670330703608685666194181234, −2.33162774070265288801488970090,
0.44481968866769282562315230742, 1.17255187624727941520423851688, 3.93882048845542461165008701717, 5.94835145095964141704986997324, 7.16846217614265535889300128780, 8.508285796533332027076334463071, 8.968219476304008451166792338977, 10.87571880940686938302772682750, 11.66792953833045278247721753239, 12.63522453054254279368954509892
