| L(s) = 1 | + (−1.67 − 0.746i)2-s + (2.15 + 0.457i)3-s + (0.912 + 1.01i)4-s + (0.0664 − 0.632i)5-s + (−3.26 − 2.37i)6-s + (−2.59 − 0.509i)7-s + (0.360 + 1.10i)8-s + (1.67 + 0.746i)9-s + (−0.582 + 1.00i)10-s + (1.50 + 2.59i)12-s + (−1.45 + 1.05i)13-s + (3.97 + 2.79i)14-s + (0.431 − 1.32i)15-s + (0.508 − 4.84i)16-s + (2.58 − 1.15i)17-s + (−2.25 − 2.50i)18-s + ⋯ |
| L(s) = 1 | + (−1.18 − 0.527i)2-s + (1.24 + 0.263i)3-s + (0.456 + 0.506i)4-s + (0.0297 − 0.282i)5-s + (−1.33 − 0.967i)6-s + (−0.981 − 0.192i)7-s + (0.127 + 0.391i)8-s + (0.558 + 0.248i)9-s + (−0.184 + 0.319i)10-s + (0.433 + 0.749i)12-s + (−0.404 + 0.293i)13-s + (1.06 + 0.745i)14-s + (0.111 − 0.343i)15-s + (0.127 − 1.21i)16-s + (0.627 − 0.279i)17-s + (−0.530 − 0.589i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.211 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.602016 - 0.746266i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.602016 - 0.746266i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.59 + 0.509i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (1.67 + 0.746i)T + (1.33 + 1.48i)T^{2} \) |
| 3 | \( 1 + (-2.15 - 0.457i)T + (2.74 + 1.22i)T^{2} \) |
| 5 | \( 1 + (-0.0664 + 0.632i)T + (-4.89 - 1.03i)T^{2} \) |
| 13 | \( 1 + (1.45 - 1.05i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-2.58 + 1.15i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-3.72 + 4.13i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (1.08 + 1.87i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.22 + 9.91i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.672 + 6.39i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (5.93 - 1.26i)T + (33.8 - 15.0i)T^{2} \) |
| 41 | \( 1 + (2.32 + 7.16i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 4.86T + 43T^{2} \) |
| 47 | \( 1 + (1.89 - 2.10i)T + (-4.91 - 46.7i)T^{2} \) |
| 53 | \( 1 + (0.780 + 7.42i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (-7.90 - 8.77i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (0.452 - 4.30i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (-0.801 + 1.38i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.47 + 2.52i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-10.7 - 11.8i)T + (-7.63 + 72.6i)T^{2} \) |
| 79 | \( 1 + (4.35 + 1.93i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (7.46 + 5.42i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (0.182 + 0.315i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.10 + 1.52i)T + (29.9 - 92.2i)T^{2} \) |
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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
−9.758047886976793688530059337187, −9.223927767032476779355222048328, −8.587172976334927968603090995452, −7.75303158618807220250469920583, −6.95036125586084600703560128497, −5.50836033040587808938060213054, −4.20592818443762973963037483921, −3.01546083796866796496548638334, −2.32231349633590770027842151565, −0.62686809633346948125053166101,
1.43555162663705160104624818849, 3.01330335716514699851253362967, 3.55394814973542444299828139247, 5.38631914883939528122056149551, 6.63806705273253594845264749082, 7.23593244964723698997018604763, 8.051276839678317003795086073255, 8.677685488738685959394691209750, 9.404079455847039880937649946613, 10.02362136579402186046203955860
