| L(s) = 1 | + (2.62 − 0.463i)2-s + (2.70 + 1.30i)3-s + (2.93 − 1.06i)4-s + (2.10 + 2.51i)5-s + (7.70 + 2.17i)6-s + (−2.69 − 4.66i)7-s + (−2.03 + 1.17i)8-s + (5.58 + 7.05i)9-s + (6.70 + 5.62i)10-s − 11.1i·11-s + (9.30 + 0.945i)12-s + (9.72 + 8.16i)13-s + (−9.24 − 11.0i)14-s + (2.41 + 9.53i)15-s + (−14.3 + 12.0i)16-s + (−15.2 − 18.2i)17-s + ⋯ |
| L(s) = 1 | + (1.31 − 0.231i)2-s + (0.900 + 0.435i)3-s + (0.732 − 0.266i)4-s + (0.421 + 0.502i)5-s + (1.28 + 0.363i)6-s + (−0.385 − 0.666i)7-s + (−0.254 + 0.146i)8-s + (0.621 + 0.783i)9-s + (0.670 + 0.562i)10-s − 1.01i·11-s + (0.775 + 0.0788i)12-s + (0.748 + 0.627i)13-s + (−0.660 − 0.787i)14-s + (0.160 + 0.635i)15-s + (−0.897 + 0.753i)16-s + (−0.898 − 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.170i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.985 - 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.51853 + 0.301537i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.51853 + 0.301537i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.70 - 1.30i)T \) |
| 19 | \( 1 + (16.8 + 8.84i)T \) |
| good | 2 | \( 1 + (-2.62 + 0.463i)T + (3.75 - 1.36i)T^{2} \) |
| 5 | \( 1 + (-2.10 - 2.51i)T + (-4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (2.69 + 4.66i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + 11.1iT - 121T^{2} \) |
| 13 | \( 1 + (-9.72 - 8.16i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (15.2 + 18.2i)T + (-50.1 + 284. i)T^{2} \) |
| 23 | \( 1 + (-13.9 - 38.1i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (16.2 + 44.7i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 - 1.25T + 961T^{2} \) |
| 37 | \( 1 - 3.67T + 1.36e3T^{2} \) |
| 41 | \( 1 + (19.2 - 3.39i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (6.18 + 2.25i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-16.9 - 46.4i)T + (-1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 + (-29.3 - 5.17i)T + (2.63e3 + 960. i)T^{2} \) |
| 59 | \( 1 + (29.1 - 80.1i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (0.669 + 0.561i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-1.49 + 8.46i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (76.8 - 13.5i)T + (4.73e3 - 1.72e3i)T^{2} \) |
| 73 | \( 1 + (-126. - 46.1i)T + (4.08e3 + 3.42e3i)T^{2} \) |
| 79 | \( 1 + (-4.76 + 4.00i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (66.7 - 38.5i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-34.3 - 94.2i)T + (-6.06e3 + 5.09e3i)T^{2} \) |
| 97 | \( 1 + (14.4 + 81.7i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−13.20267310798675116862662663730, −11.50434143108517028257280283702, −10.84392110747462442890939546586, −9.530269187557699319742704416118, −8.633634104248079881569821099411, −7.05348375986316964463750326526, −5.95439046601206913282373445482, −4.47409172818039049521610718494, −3.56009067702247158579881194013, −2.48425685611664816027652454899,
2.08876088691928657680553132061, 3.50810273534358792018816282812, 4.72843583223434374051490039983, 6.04715620661695457344037207167, 6.88297766098648118758082213367, 8.534150352361977622710210816098, 9.147139369050498935265339822488, 10.53129245911891992941106865547, 12.29947302868879575831255817041, 12.89516344600777361038358140061
