| L(s) = 1 | + (3.51 − 0.619i)2-s + (0.211 + 2.99i)3-s + (8.19 − 2.98i)4-s + (0.851 + 1.01i)5-s + (2.59 + 10.3i)6-s + (−1.32 − 2.30i)7-s + (14.5 − 8.40i)8-s + (−8.91 + 1.26i)9-s + (3.61 + 3.03i)10-s − 1.83i·11-s + (10.6 + 23.8i)12-s + (−2.55 − 2.14i)13-s + (−6.08 − 7.25i)14-s + (−2.85 + 2.76i)15-s + (19.2 − 16.1i)16-s + (−3.00 − 3.58i)17-s + ⋯ |
| L(s) = 1 | + (1.75 − 0.309i)2-s + (0.0705 + 0.997i)3-s + (2.04 − 0.745i)4-s + (0.170 + 0.203i)5-s + (0.432 + 1.72i)6-s + (−0.189 − 0.328i)7-s + (1.82 − 1.05i)8-s + (−0.990 + 0.140i)9-s + (0.361 + 0.303i)10-s − 0.166i·11-s + (0.887 + 1.98i)12-s + (−0.196 − 0.164i)13-s + (−0.434 − 0.518i)14-s + (−0.190 + 0.184i)15-s + (1.20 − 1.00i)16-s + (−0.176 − 0.210i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.74494 + 0.555945i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.74494 + 0.555945i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.211 - 2.99i)T \) |
| 19 | \( 1 + (-3.18 - 18.7i)T \) |
| good | 2 | \( 1 + (-3.51 + 0.619i)T + (3.75 - 1.36i)T^{2} \) |
| 5 | \( 1 + (-0.851 - 1.01i)T + (-4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (1.32 + 2.30i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + 1.83iT - 121T^{2} \) |
| 13 | \( 1 + (2.55 + 2.14i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (3.00 + 3.58i)T + (-50.1 + 284. i)T^{2} \) |
| 23 | \( 1 + (13.5 + 37.2i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (-13.3 - 36.6i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 - 3.76T + 961T^{2} \) |
| 37 | \( 1 + 36.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + (9.17 - 1.61i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (-5.26 - 1.91i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-1.24 - 3.42i)T + (-1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 + (10.9 + 1.93i)T + (2.63e3 + 960. i)T^{2} \) |
| 59 | \( 1 + (16.2 - 44.5i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-69.5 - 58.3i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (14.1 - 80.0i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (106. - 18.7i)T + (4.73e3 - 1.72e3i)T^{2} \) |
| 73 | \( 1 + (-126. - 46.0i)T + (4.08e3 + 3.42e3i)T^{2} \) |
| 79 | \( 1 + (-95.7 + 80.3i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (19.6 - 11.3i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-21.1 - 58.1i)T + (-6.06e3 + 5.09e3i)T^{2} \) |
| 97 | \( 1 + (-9.06 - 51.4i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−12.56203252034395520132565147043, −11.81799853235957557511937900627, −10.60006918993428771892592077742, −10.20559531312366807408180017994, −8.527187527663333618392379762197, −6.78197711786567134514817923074, −5.74058168976739431341572197069, −4.68720327887991949279577008842, −3.72229887922824616383822402619, −2.58972337500057864740730065450,
2.09863253403451038631177495837, 3.41767483894476564485916193005, 5.00476160552688907971361294569, 5.95238019117611188859691720625, 6.88605886267078735920265163479, 7.82793638448401298731399529726, 9.321987695348659452827156754793, 11.19655674301445925051808755869, 11.93676893670171850476140334754, 12.69363455181887857743862360265
