L(s) = 1 | + (1.38 + 0.268i)2-s + (1.85 + 0.746i)4-s + (3.21 + 1.16i)5-s + (0.199 + 0.0351i)7-s + (2.37 + 1.53i)8-s + (4.14 + 2.48i)10-s + (−1.37 − 3.78i)11-s + (−1.09 − 1.30i)13-s + (0.267 + 0.102i)14-s + (2.88 + 2.77i)16-s + (−1.57 + 0.908i)17-s + (−3.24 + 5.61i)19-s + (5.08 + 4.56i)20-s + (−0.894 − 5.62i)22-s + (−1.37 − 7.80i)23-s + ⋯ |
L(s) = 1 | + (0.981 + 0.190i)2-s + (0.927 + 0.373i)4-s + (1.43 + 0.523i)5-s + (0.0752 + 0.0132i)7-s + (0.839 + 0.542i)8-s + (1.31 + 0.786i)10-s + (−0.415 − 1.14i)11-s + (−0.304 − 0.362i)13-s + (0.0714 + 0.0273i)14-s + (0.721 + 0.692i)16-s + (−0.381 + 0.220i)17-s + (−0.743 + 1.28i)19-s + (1.13 + 1.02i)20-s + (−0.190 − 1.19i)22-s + (−0.286 − 1.62i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.18502 + 0.837886i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.18502 + 0.837886i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.38 - 0.268i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-3.21 - 1.16i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.199 - 0.0351i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (1.37 + 3.78i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (1.09 + 1.30i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.57 - 0.908i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.24 - 5.61i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.37 + 7.80i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (3.08 + 2.58i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (7.51 - 1.32i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (1.53 - 0.885i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.61 - 7.88i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.47 + 1.26i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.523 + 2.97i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 9.46T + 53T^{2} \) |
| 59 | \( 1 + (1.23 - 3.38i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-8.13 - 1.43i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (1.26 - 1.06i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (2.24 + 3.89i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (4.05 - 7.01i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.380 + 0.453i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-8.02 + 9.56i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (10.7 + 6.18i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (8.24 - 3.00i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.57863033554559083667773731804, −10.16531732403872108322676029717, −8.784100323237560168742792495640, −7.88247933340652672030886248316, −6.67521170192419737711566806490, −5.96557298363816876307890869212, −5.45251081629242535020971426466, −4.08839545422276861689102101870, −2.84928660362426787633709749004, −1.97509092889352629106831159149,
1.76017272432249400703861128371, 2.44248616496288329406221657703, 4.09865146918277387119048260683, 5.10520868447083563094900344911, 5.63070081176089393102925047795, 6.78598493839735736423350741947, 7.48618351563520300196945835027, 9.173173216021385835799340142222, 9.584662519558072976742304138034, 10.59786073784784978484316503430