L(s) = 1 | + (1.09 − 1.89i)2-s + (0.866 + 0.5i)3-s + (−1.39 − 2.41i)4-s + (−0.456 − 2.18i)5-s + (1.89 − 1.09i)6-s + (0.866 + 1.5i)7-s − 1.73·8-s + (−1 − 1.73i)9-s + (−4.64 − 1.52i)10-s + (−2.29 − 1.32i)11-s − 2.79i·12-s + 3.79·14-s + (0.698 − 2.12i)15-s + (0.895 − 1.55i)16-s + (3.96 − 2.29i)17-s − 4.37·18-s + ⋯ |
L(s) = 1 | + (0.773 − 1.34i)2-s + (0.499 + 0.288i)3-s + (−0.697 − 1.20i)4-s + (−0.204 − 0.978i)5-s + (0.773 − 0.446i)6-s + (0.327 + 0.566i)7-s − 0.612·8-s + (−0.333 − 0.577i)9-s + (−1.47 − 0.483i)10-s + (−0.690 − 0.398i)11-s − 0.805i·12-s + 1.01·14-s + (0.180 − 0.548i)15-s + (0.223 − 0.387i)16-s + (0.962 − 0.555i)17-s − 1.03·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.570681 - 2.43104i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.570681 - 2.43104i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.456 + 2.18i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-1.09 + 1.89i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.866 - 0.5i)T + (1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (-0.866 - 1.5i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.29 + 1.32i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-3.96 + 2.29i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.5 + 0.866i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.96 + 2.29i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.29 - 3.96i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 9.66iT - 31T^{2} \) |
| 37 | \( 1 + (-3.96 + 6.87i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.29 - 1.32i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.22 - 0.708i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 8.75T + 47T^{2} \) |
| 53 | \( 1 - 1.58iT - 53T^{2} \) |
| 59 | \( 1 + (-2.91 + 1.68i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.29 - 9.16i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7.43 - 12.8i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.08 + 1.77i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + (3.70 + 2.14i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.23 + 3.87i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.948290641351772085687645048504, −9.120061025089048224360265213850, −8.484480316459985369153407546288, −7.49660189272176201359028199042, −5.69629417688576697776855592505, −5.18821621375687590182550179120, −4.14425248824941373998145279217, −3.27815808150205333062166711289, −2.37207817523089364913397045021, −0.932315662774828692084815683020,
2.16187821691950113804367640225, 3.46118940066951583420439348054, 4.35139863487593113470909010527, 5.51526853059711874389449744948, 6.17974201070420384030811268317, 7.33125803399505085329564000670, 7.80162045114477893237338216168, 8.090329281908206116337406294670, 9.715121434912304359345983935134, 10.52868655188899379083069758917