L(s) = 1 | + (0.582 − 2.17i)2-s + (−1.71 + 0.245i)3-s + (−2.64 − 1.52i)4-s + (2.21 − 0.337i)5-s + (−0.465 + 3.86i)6-s + (−1.68 + 1.68i)8-s + (2.87 − 0.840i)9-s + (0.552 − 4.99i)10-s + (3.88 + 2.24i)11-s + (4.91 + 1.97i)12-s + (1.08 + 1.08i)13-s + (−3.70 + 1.12i)15-s + (−0.381 − 0.660i)16-s + (2.04 − 0.548i)17-s + (−0.150 − 6.74i)18-s + (3.66 − 2.11i)19-s + ⋯ |
L(s) = 1 | + (0.411 − 1.53i)2-s + (−0.989 + 0.141i)3-s + (−1.32 − 0.764i)4-s + (0.988 − 0.151i)5-s + (−0.189 + 1.57i)6-s + (−0.595 + 0.595i)8-s + (0.959 − 0.280i)9-s + (0.174 − 1.58i)10-s + (1.17 + 0.676i)11-s + (1.41 + 0.569i)12-s + (0.300 + 0.300i)13-s + (−0.957 + 0.289i)15-s + (−0.0952 − 0.165i)16-s + (0.496 − 0.133i)17-s + (−0.0354 − 1.58i)18-s + (0.839 − 0.484i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.610 + 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.610 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.763593 - 1.55268i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.763593 - 1.55268i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.71 - 0.245i)T \) |
| 5 | \( 1 + (-2.21 + 0.337i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.582 + 2.17i)T + (-1.73 - i)T^{2} \) |
| 11 | \( 1 + (-3.88 - 2.24i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.08 - 1.08i)T + 13iT^{2} \) |
| 17 | \( 1 + (-2.04 + 0.548i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-3.66 + 2.11i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.13 + 0.840i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 - 1.69T + 29T^{2} \) |
| 31 | \( 1 + (0.530 - 0.918i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.75 + 1.54i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 5.84iT - 41T^{2} \) |
| 43 | \( 1 + (-2.00 - 2.00i)T + 43iT^{2} \) |
| 47 | \( 1 + (1.36 - 5.10i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (2.23 + 8.34i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (2.35 - 4.07i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.88 + 6.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.152 + 0.569i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 4.66iT - 71T^{2} \) |
| 73 | \( 1 + (-4.22 + 1.13i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-5.78 + 3.33i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (11.0 - 11.0i)T - 83iT^{2} \) |
| 89 | \( 1 + (1.75 + 3.04i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.60 + 5.60i)T - 97iT^{2} \) |
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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
−10.18595677071522300271380792092, −9.656602013049643880116381373121, −9.011965798280584726007065445789, −7.18334603521328019787479340848, −6.29462038698732607061710282297, −5.26912422401162992074141355441, −4.50434892452275969887716586493, −3.49888751209369558908228910941, −1.99195120146078770660950770133, −1.09915173467598163268815659478,
1.42824718405595737897844123946, 3.61397784199899335201151217687, 4.82377323722243773734392855809, 5.75947989566961062915529543227, 6.08730562508107705113458783515, 6.86654036713931319466733075834, 7.75645218175924711055134100898, 8.796244526303556318416573739392, 9.743761876269962763304874596408, 10.61056084275691798501650886853