| L(s) = 1 | + (1.23 + 3.80i)2-s + (−7.28 + 5.29i)3-s + (−12.9 + 9.40i)4-s + (38.8 − 40.1i)5-s + (−29.1 − 21.1i)6-s − 37.1·7-s + (−51.7 − 37.6i)8-s + (25.0 − 77.0i)9-s + (200. + 98.0i)10-s + (90.7 + 279. i)11-s + (44.4 − 136. i)12-s + (64.7 − 199. i)13-s + (−45.8 − 141. i)14-s + (−70.1 + 498. i)15-s + (79.1 − 243. i)16-s + (408. + 296. i)17-s + ⋯ |
| L(s) = 1 | + (0.218 + 0.672i)2-s + (−0.467 + 0.339i)3-s + (−0.404 + 0.293i)4-s + (0.694 − 0.719i)5-s + (−0.330 − 0.239i)6-s − 0.286·7-s + (−0.286 − 0.207i)8-s + (0.103 − 0.317i)9-s + (0.635 + 0.310i)10-s + (0.226 + 0.696i)11-s + (0.0892 − 0.274i)12-s + (0.106 − 0.327i)13-s + (−0.0625 − 0.192i)14-s + (−0.0805 + 0.571i)15-s + (0.0772 − 0.237i)16-s + (0.342 + 0.249i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.734686374\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.734686374\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.23 - 3.80i)T \) |
| 3 | \( 1 + (7.28 - 5.29i)T \) |
| 5 | \( 1 + (-38.8 + 40.1i)T \) |
| good | 7 | \( 1 + 37.1T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-90.7 - 279. i)T + (-1.30e5 + 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-64.7 + 199. i)T + (-3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-408. - 296. i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-1.93e3 - 1.40e3i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-814. - 2.50e3i)T + (-5.20e6 + 3.78e6i)T^{2} \) |
| 29 | \( 1 + (4.97e3 - 3.61e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-999. - 726. i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (2.66e3 - 8.18e3i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (4.36e3 - 1.34e4i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 + 2.61e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-108. + 79.1i)T + (7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-2.30e4 + 1.67e4i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (366. - 1.12e3i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-7.00e3 - 2.15e4i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-4.66e4 - 3.39e4i)T + (4.17e8 + 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-6.37e4 + 4.63e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-1.05e4 - 3.23e4i)T + (-1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (6.22e4 - 4.52e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-4.11e3 - 2.98e3i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + (-2.56e4 - 7.89e4i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-3.53e4 + 2.57e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−12.58993241716374391803787402203, −11.61874772620438480974800070214, −10.00904221209142456403206206771, −9.505666189004294832677151473083, −8.206403193253823447480107390315, −6.92482292046181574134530507310, −5.70284526914934539632503330459, −5.02234368305212772938821331716, −3.55839665234600327487583074075, −1.31432300156418886216291946636,
0.64156033423651974802364924754, 2.22377045568711921046171029359, 3.48245223322208090012326417482, 5.22380282166651628917416123891, 6.22612619727359987279988245984, 7.29777225366098111780247787492, 8.990987179992865832694581704649, 9.931125696781994085612224512780, 10.97825411745462374313758621572, 11.61746603111367712659827985188
