| L(s) = 1 | + (−2.79 − 1.15i)2-s + (−0.796 + 0.158i)3-s + (3.65 + 3.65i)4-s + (−4.46 + 6.67i)5-s + (2.41 + 0.479i)6-s + (4.36 + 6.53i)7-s + (−1.34 − 3.25i)8-s + (−7.70 + 3.19i)9-s + (20.2 − 13.5i)10-s + (−1.57 + 7.92i)11-s + (−3.48 − 2.33i)12-s + (0.798 − 0.798i)13-s + (−4.63 − 23.3i)14-s + (2.49 − 6.02i)15-s − 9.98i·16-s + ⋯ |
| L(s) = 1 | + (−1.39 − 0.579i)2-s + (−0.265 + 0.0528i)3-s + (0.913 + 0.913i)4-s + (−0.892 + 1.33i)5-s + (0.402 + 0.0799i)6-s + (0.623 + 0.932i)7-s + (−0.168 − 0.407i)8-s + (−0.856 + 0.354i)9-s + (2.02 − 1.35i)10-s + (−0.143 + 0.720i)11-s + (−0.290 − 0.194i)12-s + (0.0614 − 0.0614i)13-s + (−0.331 − 1.66i)14-s + (0.166 − 0.401i)15-s − 0.624i·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0145960 - 0.208612i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0145960 - 0.208612i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + (2.79 + 1.15i)T + (2.82 + 2.82i)T^{2} \) |
| 3 | \( 1 + (0.796 - 0.158i)T + (8.31 - 3.44i)T^{2} \) |
| 5 | \( 1 + (4.46 - 6.67i)T + (-9.56 - 23.0i)T^{2} \) |
| 7 | \( 1 + (-4.36 - 6.53i)T + (-18.7 + 45.2i)T^{2} \) |
| 11 | \( 1 + (1.57 - 7.92i)T + (-111. - 46.3i)T^{2} \) |
| 13 | \( 1 + (-0.798 + 0.798i)T - 169iT^{2} \) |
| 19 | \( 1 + (-2.59 - 1.07i)T + (255. + 255. i)T^{2} \) |
| 23 | \( 1 + (8.41 + 1.67i)T + (488. + 202. i)T^{2} \) |
| 29 | \( 1 + (-2.55 - 1.70i)T + (321. + 776. i)T^{2} \) |
| 31 | \( 1 + (-2.50 - 12.5i)T + (-887. + 367. i)T^{2} \) |
| 37 | \( 1 + (-23.2 + 4.61i)T + (1.26e3 - 523. i)T^{2} \) |
| 41 | \( 1 + (12.1 + 18.1i)T + (-643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (21.4 - 8.89i)T + (1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (55.6 - 55.6i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (55.5 + 22.9i)T + (1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-10.5 - 25.3i)T + (-2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (30.3 - 20.2i)T + (1.42e3 - 3.43e3i)T^{2} \) |
| 67 | \( 1 + 117. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-103. + 20.6i)T + (4.65e3 - 1.92e3i)T^{2} \) |
| 73 | \( 1 + (-33.9 + 50.7i)T + (-2.03e3 - 4.92e3i)T^{2} \) |
| 79 | \( 1 + (18.5 - 93.4i)T + (-5.76e3 - 2.38e3i)T^{2} \) |
| 83 | \( 1 + (-45.4 + 109. i)T + (-4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-61.4 - 61.4i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (20.4 + 13.6i)T + (3.60e3 + 8.69e3i)T^{2} \) |
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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
−11.60838643830387309793731570443, −11.07403047085300517817546554734, −10.37689762658139395425060639716, −9.300159772753478629317998097467, −8.179239195445336744510151311056, −7.73078144530091292877964324424, −6.47456345815359954071334175368, −4.99887708026344159469698204002, −3.11537917122470958094187841916, −2.11952766103367931919940151540,
0.19946126401881016145193883417, 1.12396808078097483499200507046, 3.84243297488234204893262792722, 5.09075765381688734786063055305, 6.39316377331443966895129924832, 7.66303905299931568023110323759, 8.242567211206132241598017207193, 8.842650294208880117154808804062, 9.910407986336678633780603106556, 11.11266252792780097295003426820
