| L(s) = 1 | + (0.0760 + 1.73i)3-s + (−1.38 − 3.79i)5-s + (1.08 − 0.192i)7-s + (−2.98 + 0.263i)9-s + (−5.69 − 2.07i)11-s + (−2.30 − 1.93i)13-s + (6.46 − 2.68i)15-s + (−2.73 − 1.57i)17-s + (3.28 − 1.89i)19-s + (0.415 + 1.87i)21-s + (−0.347 + 1.96i)23-s + (−8.69 + 7.29i)25-s + (−0.683 − 5.15i)27-s + (2.14 + 2.56i)29-s + (3.05 + 0.538i)31-s + ⋯ |
| L(s) = 1 | + (0.0439 + 0.999i)3-s + (−0.618 − 1.69i)5-s + (0.411 − 0.0725i)7-s + (−0.996 + 0.0877i)9-s + (−1.71 − 0.624i)11-s + (−0.639 − 0.536i)13-s + (1.67 − 0.692i)15-s + (−0.663 − 0.382i)17-s + (0.753 − 0.435i)19-s + (0.0906 + 0.408i)21-s + (−0.0723 + 0.410i)23-s + (−1.73 + 1.45i)25-s + (−0.131 − 0.991i)27-s + (0.399 + 0.475i)29-s + (0.548 + 0.0967i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.399855 - 0.561753i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.399855 - 0.561753i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.0760 - 1.73i)T \) |
| good | 5 | \( 1 + (1.38 + 3.79i)T + (-3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-1.08 + 0.192i)T + (6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (5.69 + 2.07i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (2.30 + 1.93i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (2.73 + 1.57i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.28 + 1.89i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.347 - 1.96i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.14 - 2.56i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-3.05 - 0.538i)T + (29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-4.96 + 8.60i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.58 + 6.66i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (1.77 - 4.86i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.132 + 0.753i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 3.19iT - 53T^{2} \) |
| 59 | \( 1 + (-7.48 + 2.72i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (1.13 + 6.45i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (3.17 - 3.78i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (4.30 - 7.45i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.12 - 7.14i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.29 + 5.11i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (9.12 - 7.65i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-10.9 + 6.31i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (6.13 + 2.23i)T + (74.3 + 62.3i)T^{2} \) |
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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
−10.95022407133950008935731088890, −9.881945257047727827331841572193, −9.032494689554122829530994043206, −8.235715084521292977408325200319, −7.63397211383011884759086724138, −5.45392967806679208385325879053, −5.11911490608867819856004707747, −4.21004143413001615392150212629, −2.78572209053516488803841899550, −0.41000137217625594894629246921,
2.27047201022877222927516033441, 2.96961876376100852768160943072, 4.64094999859575987513968027009, 6.06868934519575632955011859574, 6.95461118730327459635483449648, 7.68188553127055596243160570229, 8.191825304916158530174039127981, 9.879459483297531773835312111439, 10.65859562152714186141204348281, 11.49578747593066544957872588035
