| L(s) = 1 | + (2.51 + 11.0i)2-s + (−37.0 − 34.3i)3-s + (−0.0173 + 0.00833i)4-s + (−172. + 26.0i)5-s + (285. − 495. i)6-s + (−201. − 348. i)7-s + (902. + 1.13e3i)8-s + (27.4 + 366. i)9-s + (−722. − 1.84e3i)10-s + (5.70e3 + 2.74e3i)11-s + (0.927 + 0.286i)12-s + (−4.06e3 + 1.03e4i)13-s + (3.34e3 − 3.09e3i)14-s + (7.30e3 + 4.97e3i)15-s + (−1.02e4 + 1.28e4i)16-s + (3.36e4 + 5.07e3i)17-s + ⋯ |
| L(s) = 1 | + (0.222 + 0.975i)2-s + (−0.792 − 0.735i)3-s + (−0.000135 + 6.50e−5i)4-s + (−0.618 + 0.0932i)5-s + (0.540 − 0.936i)6-s + (−0.221 − 0.384i)7-s + (0.623 + 0.781i)8-s + (0.0125 + 0.167i)9-s + (−0.228 − 0.582i)10-s + (1.29 + 0.622i)11-s + (0.000154 + 4.77e−5i)12-s + (−0.512 + 1.30i)13-s + (0.325 − 0.301i)14-s + (0.558 + 0.380i)15-s + (−0.623 + 0.781i)16-s + (1.66 + 0.250i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0419 - 0.999i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.0419 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.06713 + 1.02326i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06713 + 1.02326i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-5.19e5 + 4.00e4i)T \) |
| good | 2 | \( 1 + (-2.51 - 11.0i)T + (-115. + 55.5i)T^{2} \) |
| 3 | \( 1 + (37.0 + 34.3i)T + (163. + 2.18e3i)T^{2} \) |
| 5 | \( 1 + (172. - 26.0i)T + (7.46e4 - 2.30e4i)T^{2} \) |
| 7 | \( 1 + (201. + 348. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-5.70e3 - 2.74e3i)T + (1.21e7 + 1.52e7i)T^{2} \) |
| 13 | \( 1 + (4.06e3 - 1.03e4i)T + (-4.59e7 - 4.26e7i)T^{2} \) |
| 17 | \( 1 + (-3.36e4 - 5.07e3i)T + (3.92e8 + 1.20e8i)T^{2} \) |
| 19 | \( 1 + (183. - 2.44e3i)T + (-8.83e8 - 1.33e8i)T^{2} \) |
| 23 | \( 1 + (5.97e4 - 4.07e4i)T + (1.24e9 - 3.16e9i)T^{2} \) |
| 29 | \( 1 + (-1.28e5 + 1.18e5i)T + (1.28e9 - 1.72e10i)T^{2} \) |
| 31 | \( 1 + (-3.04e5 - 9.39e4i)T + (2.27e10 + 1.54e10i)T^{2} \) |
| 37 | \( 1 + (-4.20e4 + 7.28e4i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-9.49e4 - 4.15e5i)T + (-1.75e11 + 8.45e10i)T^{2} \) |
| 47 | \( 1 + (2.77e5 - 1.33e5i)T + (3.15e11 - 3.96e11i)T^{2} \) |
| 53 | \( 1 + (3.54e5 + 9.02e5i)T + (-8.61e11 + 7.99e11i)T^{2} \) |
| 59 | \( 1 + (1.08e6 - 1.36e6i)T + (-5.53e11 - 2.42e12i)T^{2} \) |
| 61 | \( 1 + (6.37e5 - 1.96e5i)T + (2.59e12 - 1.77e12i)T^{2} \) |
| 67 | \( 1 + (-3.03e4 + 4.05e5i)T + (-5.99e12 - 9.03e11i)T^{2} \) |
| 71 | \( 1 + (2.01e6 + 1.37e6i)T + (3.32e12 + 8.46e12i)T^{2} \) |
| 73 | \( 1 + (2.01e6 - 5.13e6i)T + (-8.09e12 - 7.51e12i)T^{2} \) |
| 79 | \( 1 + (-6.57e5 - 1.13e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (3.09e6 + 2.87e6i)T + (2.02e12 + 2.70e13i)T^{2} \) |
| 89 | \( 1 + (-1.35e5 - 1.25e5i)T + (3.30e12 + 4.41e13i)T^{2} \) |
| 97 | \( 1 + (-9.92e6 - 4.77e6i)T + (5.03e13 + 6.31e13i)T^{2} \) |
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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
−14.72103204076905494579195175269, −13.89835255591027479051380204687, −11.98027181199803719830653028143, −11.76257467831522450554720623487, −9.840309810899778401378219443651, −7.81033312094577225052689755779, −6.87207961910895741656457030724, −6.04421693532605399926316300115, −4.24571963165077386678813750489, −1.37103418015001746882707565648,
0.72740633943407697382525745874, 3.05825447445172824356798241760, 4.36538975972547262998767194022, 5.97709135865111973635221669090, 7.905947541024177503955819767943, 9.836801719382972688178229787999, 10.67839664231216117605634455736, 11.96191975127680359640254015259, 12.21256284629482256480614442321, 14.04929583783119660466395394607
