| L(s) = 1 | + (0.682 + 1.59i)2-s + (−3.05 + 0.554i)3-s + (−0.698 + 0.730i)4-s + (0.984 + 3.03i)5-s + (−2.96 − 4.49i)6-s + (−1.95 − 1.70i)7-s + (1.60 + 0.602i)8-s + (6.21 − 2.33i)9-s + (−4.16 + 3.63i)10-s + (2.87 + 0.793i)11-s + (1.72 − 2.61i)12-s + (1.82 − 0.502i)13-s + (1.39 − 4.28i)14-s + (−4.68 − 8.71i)15-s + (0.224 + 4.99i)16-s + (−4.31 − 3.13i)17-s + ⋯ |
| L(s) = 1 | + (0.482 + 1.12i)2-s + (−1.76 + 0.320i)3-s + (−0.349 + 0.365i)4-s + (0.440 + 1.35i)5-s + (−1.21 − 1.83i)6-s + (−0.739 − 0.645i)7-s + (0.567 + 0.213i)8-s + (2.07 − 0.777i)9-s + (−1.31 + 1.15i)10-s + (0.867 + 0.239i)11-s + (0.499 − 0.756i)12-s + (0.505 − 0.139i)13-s + (0.372 − 1.14i)14-s + (−1.21 − 2.24i)15-s + (0.0560 + 1.24i)16-s + (−1.04 − 0.760i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.539 - 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.539 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.378666 + 0.692455i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.378666 + 0.692455i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 + (3.28 - 7.75i)T \) |
| good | 2 | \( 1 + (-0.682 - 1.59i)T + (-1.38 + 1.44i)T^{2} \) |
| 3 | \( 1 + (3.05 - 0.554i)T + (2.80 - 1.05i)T^{2} \) |
| 5 | \( 1 + (-0.984 - 3.03i)T + (-4.04 + 2.93i)T^{2} \) |
| 7 | \( 1 + (1.95 + 1.70i)T + (0.939 + 6.93i)T^{2} \) |
| 11 | \( 1 + (-2.87 - 0.793i)T + (9.44 + 5.64i)T^{2} \) |
| 13 | \( 1 + (-1.82 + 0.502i)T + (11.1 - 6.66i)T^{2} \) |
| 17 | \( 1 + (4.31 + 3.13i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.616 + 1.14i)T + (-10.4 - 15.8i)T^{2} \) |
| 23 | \( 1 + (-0.0673 - 0.295i)T + (-20.7 + 9.97i)T^{2} \) |
| 29 | \( 1 + (-0.418 + 3.08i)T + (-27.9 - 7.71i)T^{2} \) |
| 31 | \( 1 + (-0.0495 + 1.10i)T + (-30.8 - 2.77i)T^{2} \) |
| 37 | \( 1 + (-0.852 + 3.73i)T + (-33.3 - 16.0i)T^{2} \) |
| 41 | \( 1 + (1.29 - 1.62i)T + (-9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (4.77 - 0.430i)T + (42.3 - 7.67i)T^{2} \) |
| 47 | \( 1 + (-8.29 - 1.50i)T + (44.0 + 16.5i)T^{2} \) |
| 53 | \( 1 + (-0.272 - 0.284i)T + (-2.37 + 52.9i)T^{2} \) |
| 59 | \( 1 + (-6.26 + 9.49i)T + (-23.1 - 54.2i)T^{2} \) |
| 61 | \( 1 + (5.47 - 4.77i)T + (8.18 - 60.4i)T^{2} \) |
| 67 | \( 1 + (-7.33 + 7.67i)T + (-3.00 - 66.9i)T^{2} \) |
| 73 | \( 1 + (-2.65 - 6.21i)T + (-50.4 + 52.7i)T^{2} \) |
| 79 | \( 1 + (16.0 + 6.02i)T + (59.4 + 51.9i)T^{2} \) |
| 83 | \( 1 + (7.68 - 11.6i)T + (-32.6 - 76.3i)T^{2} \) |
| 89 | \( 1 + (6.19 + 6.48i)T + (-3.99 + 88.9i)T^{2} \) |
| 97 | \( 1 + (2.79 + 3.50i)T + (-21.5 + 94.5i)T^{2} \) |
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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
−15.34550595233344891078392103034, −14.12757347533295857078851704142, −13.13831647659824056041278537897, −11.48095814554554372157261941495, −10.75712709527874212681501112293, −9.795388260562414179151883405589, −6.96474048691061234631579920818, −6.69892975126914194646276065621, −5.73152630997752492184137745810, −4.20806712963640638181319480967,
1.42093127401725229603429736997, 4.27737358063436367096794841445, 5.52711610396771675050953773382, 6.60132530636717261903729073641, 8.940057675512750869072820934539, 10.25398010640687705990271617824, 11.37704973864026064018913671046, 12.17482185464864676098480767168, 12.76326001642689879916097705871, 13.48389705719659755684073436548
