| L(s) = 1 | + (−0.959 − 0.281i)2-s + (−0.749 − 0.649i)3-s + (0.841 + 0.540i)4-s + (−0.540 + 3.76i)5-s + (0.536 + 0.834i)6-s + (−2.61 + 0.378i)7-s + (−0.654 − 0.755i)8-s + (−0.286 − 1.99i)9-s + (1.57 − 3.45i)10-s + (−1.64 − 5.59i)11-s + (−0.279 − 0.951i)12-s + (0.519 + 0.237i)13-s + (2.61 + 0.374i)14-s + (2.84 − 2.46i)15-s + (0.415 + 0.909i)16-s + (5.39 − 3.46i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 − 0.199i)2-s + (−0.432 − 0.375i)3-s + (0.420 + 0.270i)4-s + (−0.241 + 1.68i)5-s + (0.218 + 0.340i)6-s + (−0.989 + 0.143i)7-s + (−0.231 − 0.267i)8-s + (−0.0956 − 0.665i)9-s + (0.499 − 1.09i)10-s + (−0.495 − 1.68i)11-s + (−0.0806 − 0.274i)12-s + (0.144 + 0.0658i)13-s + (0.699 + 0.100i)14-s + (0.735 − 0.637i)15-s + (0.103 + 0.227i)16-s + (1.30 − 0.841i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.666 + 0.745i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.128381 - 0.287061i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.128381 - 0.287061i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 + 0.281i)T \) |
| 7 | \( 1 + (2.61 - 0.378i)T \) |
| 23 | \( 1 + (-2.13 + 4.29i)T \) |
| good | 3 | \( 1 + (0.749 + 0.649i)T + (0.426 + 2.96i)T^{2} \) |
| 5 | \( 1 + (0.540 - 3.76i)T + (-4.79 - 1.40i)T^{2} \) |
| 11 | \( 1 + (1.64 + 5.59i)T + (-9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-0.519 - 0.237i)T + (8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-5.39 + 3.46i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (5.54 + 3.56i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-0.651 + 0.418i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (5.47 - 4.74i)T + (4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (0.936 - 0.134i)T + (35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (2.88 + 0.415i)T + (39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-1.26 - 1.10i)T + (6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 1.08iT - 47T^{2} \) |
| 53 | \( 1 + (3.78 - 1.72i)T + (34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (8.99 + 4.10i)T + (38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (4.78 + 5.52i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (1.75 - 5.98i)T + (-56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-9.07 - 2.66i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (2.13 - 3.31i)T + (-30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (3.57 + 1.63i)T + (51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (1.38 + 9.63i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-5.73 + 6.62i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (0.176 - 1.22i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−10.98972604279547195175122805227, −10.66419367625383731649872367203, −9.515910846537483649425181516139, −8.491078873309689246371521149851, −7.22723652171525992473245904865, −6.55944003060731314782070269702, −5.86487141501043878061786449261, −3.35640883046443745990323729554, −2.90199930654795190143217664036, −0.28493394778369751797263518081,
1.74438979389488063674637427696, 4.00671596997136985062404372915, 5.09395230606645246415857302701, 5.94682832102894729689673801518, 7.49085026046423784771160949543, 8.167815953885235318445418824886, 9.315515311152786121530501230073, 9.960430900961108364159972678721, 10.72405488074438484112603995384, 12.19874477888736549873155994666
