| L(s) = 1 | + (2 − 3.46i)2-s + (−14.6 − 5.19i)3-s + (−7.99 − 13.8i)4-s + (−28.1 − 48.8i)5-s + (−47.3 + 40.5i)6-s + (11.1 − 19.3i)7-s − 63.9·8-s + (189 + 152. i)9-s − 225.·10-s + (296. − 513. i)11-s + (45.5 + 245. i)12-s + (129. + 224. i)13-s + (−44.6 − 77.2i)14-s + (160. + 864. i)15-s + (−128 + 221. i)16-s + 1.62e3·17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.942 − 0.333i)3-s + (−0.249 − 0.433i)4-s + (−0.504 − 0.873i)5-s + (−0.537 + 0.459i)6-s + (0.0860 − 0.148i)7-s − 0.353·8-s + (0.777 + 0.628i)9-s − 0.713·10-s + (0.738 − 1.27i)11-s + (0.0913 + 0.491i)12-s + (0.212 + 0.368i)13-s + (−0.0608 − 0.105i)14-s + (0.184 + 0.991i)15-s + (−0.125 + 0.216i)16-s + 1.36·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.358642 - 0.957765i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.358642 - 0.957765i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2 + 3.46i)T \) |
| 3 | \( 1 + (14.6 + 5.19i)T \) |
| good | 5 | \( 1 + (28.1 + 48.8i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-11.1 + 19.3i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-296. + 513. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-129. - 224. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 - 1.62e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.64e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (1.56e3 + 2.71e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (62.2 - 107. i)T + (-1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 + (-4.37e3 - 7.57e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 - 9.74e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-3.73e3 - 6.47e3i)T + (-5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-6.25e3 + 1.08e4i)T + (-7.35e7 - 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-4.40e3 + 7.63e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + 1.07e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-7.77e3 - 1.34e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-5.96e3 + 1.03e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (3.08e3 + 5.35e3i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + 1.04e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.83e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (8.92e3 - 1.54e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-2.16e4 + 3.74e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + 9.40e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-4.07e4 + 7.05e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−16.98047680108832992097513354983, −16.21422977660534182139442444076, −14.18963639247044396136160532301, −12.69221572029100091838100060267, −11.83865182996358570874042275611, −10.57559409097599858330115099374, −8.486321866727081155140317189704, −6.13022856964328002006875292662, −4.34834546863170708381835743615, −0.874351368839317877182743688301,
4.13378035640076703048240740136, 6.08349614267157565472955749215, 7.48744271895944263411487098725, 9.893578548057804709635961400549, 11.45150232959813451227415718304, 12.62411716956373630162670812521, 14.71215661462253925266403506559, 15.37356826798086773340906034448, 16.85083147702780095971127397093, 17.80723027863276360141257158981
