L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (0.725 − 0.418i)5-s + 0.999·6-s + (−2.44 + 1.02i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−0.418 + 0.725i)10-s + (−3.15 − 1.01i)11-s + (−0.866 + 0.499i)12-s + 2.59·13-s + (1.60 − 2.10i)14-s − 0.837·15-s + (−0.5 − 0.866i)16-s + (−2.98 + 5.17i)17-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (0.324 − 0.187i)5-s + 0.408·6-s + (−0.922 + 0.386i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.132 + 0.229i)10-s + (−0.951 − 0.306i)11-s + (−0.249 + 0.144i)12-s + 0.719·13-s + (0.428 − 0.562i)14-s − 0.216·15-s + (−0.125 − 0.216i)16-s + (−0.724 + 1.25i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.670 - 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.670 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.159234 + 0.358668i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.159234 + 0.358668i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 7 | \( 1 + (2.44 - 1.02i)T \) |
| 11 | \( 1 + (3.15 + 1.01i)T \) |
good | 5 | \( 1 + (-0.725 + 0.418i)T + (2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 - 2.59T + 13T^{2} \) |
| 17 | \( 1 + (2.98 - 5.17i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.55 - 2.69i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.43 + 2.48i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 5.38iT - 29T^{2} \) |
| 31 | \( 1 + (0.913 + 0.527i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.49 - 9.51i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 + 1.27iT - 43T^{2} \) |
| 47 | \( 1 + (10.6 - 6.12i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.58 - 4.48i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (8.38 + 4.84i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.03 + 3.52i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.51 - 11.2i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 + (-4.95 + 8.58i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-11.7 + 6.75i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 2.99T + 83T^{2} \) |
| 89 | \( 1 + (-7.28 + 4.20i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 0.786iT - 97T^{2} \) |
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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
−11.14267354231440052726159185077, −10.42273819408512151977679512653, −9.616854795064716382303162445066, −8.588553073231831682025354439875, −7.87056803954128826266946248160, −6.50477503370422852467505985095, −6.08707655343468923103196088214, −5.04207212894550137032625192358, −3.30040297975469260744357945625, −1.68395959354423926997103661904,
0.30251181642814331507091943220, 2.39165465111588241835397365918, 3.61239914349499310135602800245, 4.94576909095005004517185577590, 6.18858927099357310234017813668, 7.00612214715871424965889008258, 8.019556177040963341138416460517, 9.344893382220390019884862746613, 9.789785081497822393219453383406, 10.66509707840222996447491984809