| L(s) = 1 | + (−2.37 − 1.18i)2-s + (−1.03 + 2.34i)3-s + (3.03 + 4.02i)4-s + (0.253 − 0.173i)5-s + (5.22 − 4.33i)6-s + (3.33 + 0.154i)7-s + (−1.48 − 7.93i)8-s + (−2.39 − 2.62i)9-s + (−0.807 + 0.112i)10-s + (0.0907 − 0.650i)11-s + (−12.5 + 2.95i)12-s + (2.39 − 0.447i)13-s + (−7.73 − 4.30i)14-s + (0.144 + 0.773i)15-s + (−3.09 + 10.8i)16-s + (1.76 + 3.72i)17-s + ⋯ |
| L(s) = 1 | + (−1.67 − 0.836i)2-s + (−0.597 + 1.35i)3-s + (1.51 + 2.01i)4-s + (0.113 − 0.0776i)5-s + (2.13 − 1.77i)6-s + (1.25 + 0.0582i)7-s + (−0.524 − 2.80i)8-s + (−0.798 − 0.875i)9-s + (−0.255 + 0.0356i)10-s + (0.0273 − 0.196i)11-s + (−3.62 + 0.852i)12-s + (0.663 − 0.123i)13-s + (−2.06 − 1.15i)14-s + (0.0373 + 0.199i)15-s + (−0.774 + 2.72i)16-s + (0.427 + 0.903i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.679 - 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.528531 + 0.230968i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.528531 + 0.230968i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-1.76 - 3.72i)T \) |
| good | 2 | \( 1 + (2.37 + 1.18i)T + (1.20 + 1.59i)T^{2} \) |
| 3 | \( 1 + (1.03 - 2.34i)T + (-2.02 - 2.21i)T^{2} \) |
| 5 | \( 1 + (-0.253 + 0.173i)T + (1.80 - 4.66i)T^{2} \) |
| 7 | \( 1 + (-3.33 - 0.154i)T + (6.97 + 0.645i)T^{2} \) |
| 11 | \( 1 + (-0.0907 + 0.650i)T + (-10.5 - 3.01i)T^{2} \) |
| 13 | \( 1 + (-2.39 + 0.447i)T + (12.1 - 4.69i)T^{2} \) |
| 19 | \( 1 + (-5.77 + 2.87i)T + (11.4 - 15.1i)T^{2} \) |
| 23 | \( 1 + (5.57 + 0.257i)T + (22.9 + 2.12i)T^{2} \) |
| 29 | \( 1 + (-7.55 - 1.05i)T + (27.8 + 7.93i)T^{2} \) |
| 31 | \( 1 + (-2.26 + 3.30i)T + (-11.1 - 28.9i)T^{2} \) |
| 37 | \( 1 + (0.719 - 3.05i)T + (-33.1 - 16.4i)T^{2} \) |
| 41 | \( 1 + (-4.92 - 2.17i)T + (27.6 + 30.2i)T^{2} \) |
| 43 | \( 1 + (11.3 - 3.23i)T + (36.5 - 22.6i)T^{2} \) |
| 47 | \( 1 + (-2.60 - 2.37i)T + (4.33 + 46.7i)T^{2} \) |
| 53 | \( 1 + (-5.54 - 6.07i)T + (-4.89 + 52.7i)T^{2} \) |
| 59 | \( 1 + (9.58 - 0.887i)T + (57.9 - 10.8i)T^{2} \) |
| 61 | \( 1 + (7.55 - 6.27i)T + (11.2 - 59.9i)T^{2} \) |
| 67 | \( 1 + (-6.03 - 12.1i)T + (-40.3 + 53.4i)T^{2} \) |
| 71 | \( 1 + (-0.403 + 8.71i)T + (-70.6 - 6.55i)T^{2} \) |
| 73 | \( 1 + (6.74 + 12.1i)T + (-38.4 + 62.0i)T^{2} \) |
| 79 | \( 1 + (-0.846 + 2.52i)T + (-63.0 - 47.6i)T^{2} \) |
| 83 | \( 1 + (-1.82 + 4.70i)T + (-61.3 - 55.9i)T^{2} \) |
| 89 | \( 1 + (-6.94 - 1.29i)T + (82.9 + 32.1i)T^{2} \) |
| 97 | \( 1 + (-0.430 + 9.30i)T + (-96.5 - 8.95i)T^{2} \) |
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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
−11.53859889543733449840478633417, −10.75629375362975593989906555364, −10.16669321524375157596659840847, −9.335539297864021169209173418343, −8.426574626076920141262882404262, −7.66031233783575038026011701017, −5.93879129435204493287147290243, −4.55005099158281470529308580835, −3.25269547359679437418867706574, −1.39949175309018248492061438237,
0.955503395549310816783603579805, 1.95709212502318078668791661550, 5.24821983004095303692105823034, 6.18582483827812498945776483066, 7.04727347838484990070646646442, 7.901055026172754713447418367045, 8.317023914108553692870810725593, 9.693658551435467939558700521608, 10.60009759462620081046523597471, 11.68827515497442618108491138744
