| L(s) = 1 | + (−0.173 + 0.984i)2-s + (0.766 + 0.642i)3-s + (−0.939 − 0.342i)4-s + (−3.20 + 1.16i)5-s + (−0.766 + 0.642i)6-s + (2.43 + 4.22i)7-s + (0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.592 − 3.35i)10-s + (1.70 − 2.95i)11-s + (−0.499 − 0.866i)12-s + (2.08 − 1.74i)13-s + (−4.58 + 1.66i)14-s + (−3.20 − 1.16i)15-s + (0.766 + 0.642i)16-s + (0.205 − 1.16i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 + 0.696i)2-s + (0.442 + 0.371i)3-s + (−0.469 − 0.171i)4-s + (−1.43 + 0.521i)5-s + (−0.312 + 0.262i)6-s + (0.922 + 1.59i)7-s + (0.176 − 0.306i)8-s + (0.0578 + 0.328i)9-s + (−0.187 − 1.06i)10-s + (0.514 − 0.890i)11-s + (−0.144 − 0.249i)12-s + (0.577 − 0.484i)13-s + (−1.22 + 0.446i)14-s + (−0.827 − 0.301i)15-s + (0.191 + 0.160i)16-s + (0.0498 − 0.282i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.586607 + 0.756639i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.586607 + 0.756639i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 3 | \( 1 + (-0.766 - 0.642i)T \) |
| 19 | \( 1 + (2.52 + 3.55i)T \) |
| good | 5 | \( 1 + (3.20 - 1.16i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-2.43 - 4.22i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.70 + 2.95i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.08 + 1.74i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.205 + 1.16i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-3.20 - 1.16i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.655 - 3.71i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (3.30 + 5.72i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 6.75T + 37T^{2} \) |
| 41 | \( 1 + (-5.02 - 4.21i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (3.91 - 1.42i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.496 - 2.81i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (0.592 + 0.215i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-2.02 + 11.4i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (6.48 + 2.36i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.123 - 0.698i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (7.47 - 2.72i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (9.76 + 8.19i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-0.228 - 0.191i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (7.80 + 13.5i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.18 + 4.35i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (1.23 - 7.01i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−14.44401953652439731138519189517, −12.93758482200047780057198275397, −11.51213024797225160127611878864, −11.08673402819802637297261717670, −9.121162578854995786795615336701, −8.452388261204421262131609389268, −7.61847216762413594311916299717, −6.03434357334011024512996321350, −4.63884173917082172136083794530, −3.13064920329042396182175552699,
1.31991710745951298477186348114, 3.90842535542507025460507073959, 4.40107894957863005498224202632, 7.11096465428150460927484602150, 7.912340443219674967865210591439, 8.817673227052041354022286560938, 10.36075747036530627533833028600, 11.31380374474634351098479675715, 12.16659199063139935051061021528, 13.12713107254123563426333179780
