L(s) = 1 | + (0.686 − 0.727i)2-s + (1.65 − 0.524i)3-s + (−0.0581 − 0.998i)4-s + (−1.97 − 0.230i)5-s + (0.750 − 1.56i)6-s + (0.868 − 0.436i)7-s + (−0.766 − 0.642i)8-s + (2.44 − 1.73i)9-s + (−1.52 + 1.27i)10-s + (−1.12 + 2.61i)11-s + (−0.619 − 1.61i)12-s + (1.19 − 0.284i)13-s + (0.278 − 0.931i)14-s + (−3.37 + 0.654i)15-s + (−0.993 + 0.116i)16-s + (0.944 + 5.35i)17-s + ⋯ |
L(s) = 1 | + (0.485 − 0.514i)2-s + (0.952 − 0.303i)3-s + (−0.0290 − 0.499i)4-s + (−0.881 − 0.103i)5-s + (0.306 − 0.637i)6-s + (0.328 − 0.164i)7-s + (−0.270 − 0.227i)8-s + (0.816 − 0.577i)9-s + (−0.480 + 0.403i)10-s + (−0.340 + 0.789i)11-s + (−0.178 − 0.466i)12-s + (0.332 − 0.0788i)13-s + (0.0745 − 0.248i)14-s + (−0.871 + 0.168i)15-s + (−0.248 + 0.0290i)16-s + (0.229 + 1.29i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.491 + 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.491 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.45162 - 0.847406i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.45162 - 0.847406i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.686 + 0.727i)T \) |
| 3 | \( 1 + (-1.65 + 0.524i)T \) |
good | 5 | \( 1 + (1.97 + 0.230i)T + (4.86 + 1.15i)T^{2} \) |
| 7 | \( 1 + (-0.868 + 0.436i)T + (4.18 - 5.61i)T^{2} \) |
| 11 | \( 1 + (1.12 - 2.61i)T + (-7.54 - 8.00i)T^{2} \) |
| 13 | \( 1 + (-1.19 + 0.284i)T + (11.6 - 5.83i)T^{2} \) |
| 17 | \( 1 + (-0.944 - 5.35i)T + (-15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (0.724 - 4.10i)T + (-17.8 - 6.49i)T^{2} \) |
| 23 | \( 1 + (5.83 + 2.92i)T + (13.7 + 18.4i)T^{2} \) |
| 29 | \( 1 + (-1.00 - 3.37i)T + (-24.2 + 15.9i)T^{2} \) |
| 31 | \( 1 + (-7.08 + 4.66i)T + (12.2 - 28.4i)T^{2} \) |
| 37 | \( 1 + (3.74 + 1.36i)T + (28.3 + 23.7i)T^{2} \) |
| 41 | \( 1 + (6.52 + 6.91i)T + (-2.38 + 40.9i)T^{2} \) |
| 43 | \( 1 + (-6.25 - 8.40i)T + (-12.3 + 41.1i)T^{2} \) |
| 47 | \( 1 + (3.54 + 2.33i)T + (18.6 + 43.1i)T^{2} \) |
| 53 | \( 1 + (-0.380 - 0.659i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.0874 + 0.202i)T + (-40.4 + 42.9i)T^{2} \) |
| 61 | \( 1 + (-0.258 + 4.44i)T + (-60.5 - 7.08i)T^{2} \) |
| 67 | \( 1 + (-2.08 + 6.95i)T + (-55.9 - 36.8i)T^{2} \) |
| 71 | \( 1 + (-11.3 + 9.56i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (5.51 + 4.63i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (1.53 - 1.62i)T + (-4.59 - 78.8i)T^{2} \) |
| 83 | \( 1 + (-8.03 + 8.52i)T + (-4.82 - 82.8i)T^{2} \) |
| 89 | \( 1 + (-0.108 - 0.0910i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (10.5 - 1.23i)T + (94.3 - 22.3i)T^{2} \) |
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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
−12.51499996975749255213638294960, −12.13382643755425537118441060096, −10.68432217331025230928636191156, −9.823031135869409313948021757264, −8.327709574628478996488534233037, −7.78612259382889194701970425405, −6.29207584689613414820938105187, −4.44087754689297421693581637800, −3.59764613107670411759621143482, −1.89490617403707246793688727530,
2.85980349738214025919337645690, 4.01089878708677680260998672117, 5.19798498205092344202966631134, 6.88750254760227010162326521078, 7.968118847566985557649500692504, 8.554184953732941809888324645686, 9.850410113339606625143391998423, 11.25674585865913416248341347014, 12.04496513728150456248498628009, 13.53935094769532686268670228763