| L(s) = 1 | + (1.42 − 1.19i)2-s + (−1.62 − 0.610i)3-s + (0.251 − 1.42i)4-s + (0.629 − 1.73i)5-s + (−3.03 + 1.06i)6-s + (−1.66 − 2.88i)7-s + (0.510 + 0.884i)8-s + (2.25 + 1.98i)9-s + (−1.16 − 3.21i)10-s − 2.01i·11-s + (−1.28 + 2.16i)12-s + (−0.355 − 0.975i)13-s + (−5.81 − 2.11i)14-s + (−2.07 + 2.41i)15-s + (4.50 + 1.64i)16-s + (−2.49 + 6.85i)17-s + ⋯ |
| L(s) = 1 | + (1.00 − 0.844i)2-s + (−0.935 − 0.352i)3-s + (0.125 − 0.714i)4-s + (0.281 − 0.773i)5-s + (−1.23 + 0.435i)6-s + (−0.629 − 1.09i)7-s + (0.180 + 0.312i)8-s + (0.751 + 0.660i)9-s + (−0.369 − 1.01i)10-s − 0.606i·11-s + (−0.369 + 0.623i)12-s + (−0.0984 − 0.270i)13-s + (−1.55 − 0.565i)14-s + (−0.536 + 0.624i)15-s + (1.12 + 0.410i)16-s + (−0.605 + 1.66i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.377 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.801785 - 1.19320i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.801785 - 1.19320i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.62 + 0.610i)T \) |
| 19 | \( 1 + (-4.33 + 0.456i)T \) |
| good | 2 | \( 1 + (-1.42 + 1.19i)T + (0.347 - 1.96i)T^{2} \) |
| 5 | \( 1 + (-0.629 + 1.73i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (1.66 + 2.88i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + 2.01iT - 11T^{2} \) |
| 13 | \( 1 + (0.355 + 0.975i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (2.49 - 6.85i)T + (-13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-5.68 - 1.00i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.250 - 1.41i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + 3.08iT - 31T^{2} \) |
| 37 | \( 1 - 2.56iT - 37T^{2} \) |
| 41 | \( 1 + (6.61 - 5.55i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (0.553 + 3.13i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (8.65 + 1.52i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (4.17 + 3.50i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-1.40 - 7.94i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-10.4 + 3.81i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (5.53 - 6.59i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-4.33 + 3.63i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (2.19 + 12.4i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (3.44 - 9.47i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (7.44 - 4.29i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (1.35 - 7.67i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (10.2 + 12.1i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−12.75011864502032768407732698525, −11.55337271698283342133322495216, −10.82799260458991821226816845352, −9.941002007952839287964025958349, −8.286399638294265584764847600022, −6.87902464401351775685057127065, −5.64839545945355558076618625136, −4.68821420368976584211772876778, −3.46947305492832196078774292880, −1.31734728151487119348692282212,
3.05294130874989378854565059826, 4.76004211868876362348252760684, 5.50617659430040670672422322660, 6.65039632809278497922982413548, 7.07852431543999736440642433987, 9.279180989235238060275160262664, 10.03353426212754686851552612022, 11.33835751436804829578047856613, 12.23191762365331102722569263674, 13.09470184098239664347383109900
