| L(s) = 1 | + (0.415 + 0.909i)2-s + (−0.959 + 0.281i)3-s + (−0.654 + 0.755i)4-s + (1.27 + 0.818i)5-s + (−0.654 − 0.755i)6-s + (0.369 + 2.56i)7-s + (−0.959 − 0.281i)8-s + (0.841 − 0.540i)9-s + (−0.215 + 1.49i)10-s + (−2.08 + 4.55i)11-s + (0.415 − 0.909i)12-s + (0.686 − 4.77i)13-s + (−2.18 + 1.40i)14-s + (−1.45 − 0.426i)15-s + (−0.142 − 0.989i)16-s + (0.565 + 0.652i)17-s + ⋯ |
| L(s) = 1 | + (0.293 + 0.643i)2-s + (−0.553 + 0.162i)3-s + (−0.327 + 0.377i)4-s + (0.569 + 0.365i)5-s + (−0.267 − 0.308i)6-s + (0.139 + 0.970i)7-s + (−0.339 − 0.0996i)8-s + (0.280 − 0.180i)9-s + (−0.0681 + 0.473i)10-s + (−0.627 + 1.37i)11-s + (0.119 − 0.262i)12-s + (0.190 − 1.32i)13-s + (−0.583 + 0.374i)14-s + (−0.374 − 0.110i)15-s + (−0.0355 − 0.247i)16-s + (0.137 + 0.158i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0659 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0659 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.759704 + 0.811544i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.759704 + 0.811544i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.415 - 0.909i)T \) |
| 3 | \( 1 + (0.959 - 0.281i)T \) |
| 23 | \( 1 + (-4.66 + 1.11i)T \) |
| good | 5 | \( 1 + (-1.27 - 0.818i)T + (2.07 + 4.54i)T^{2} \) |
| 7 | \( 1 + (-0.369 - 2.56i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (2.08 - 4.55i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.686 + 4.77i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (-0.565 - 0.652i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (-4.58 + 5.29i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (2.03 + 2.35i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-1.12 - 0.330i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (0.506 - 0.325i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (-4.45 - 2.86i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (1.40 - 0.411i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + 12.6T + 47T^{2} \) |
| 53 | \( 1 + (0.602 + 4.18i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (0.566 - 3.94i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (-6.96 - 2.04i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (6.70 + 14.6i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (-2.44 - 5.35i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (9.41 - 10.8i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (1.47 - 10.2i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (-2.22 + 1.42i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (-5.02 + 1.47i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (-12.5 - 8.08i)T + (40.2 + 88.2i)T^{2} \) |
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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
−13.29969864976203297311579499569, −12.65452250442538249898145199393, −11.54453704344056500186068616707, −10.27598246080455462440630645223, −9.387879242997334596795153238783, −7.972715230205607452856964237100, −6.81975451884528551863039974859, −5.61238979053249177866476391983, −4.88932216475649440393934797356, −2.75047125335412103180190272868,
1.34455069556571945032477136924, 3.54561354565591521200043278808, 5.03237322461664153753809636446, 6.05578671160112975376319619338, 7.49874969694200816749646595388, 8.984840362768920975816447657571, 10.07857919916091534743409205914, 11.05452605098036511473696734745, 11.74248937653439769776916224076, 13.07767776224235039230057138604
