| L(s) = 1 | + (−1.27 + 0.621i)2-s + (−2.41 + 0.272i)3-s + (1.22 − 1.57i)4-s + (0.786 − 1.63i)5-s + (2.89 − 1.84i)6-s + (2.13 + 1.70i)7-s + (−0.575 + 2.76i)8-s + (2.83 − 0.647i)9-s + (0.0166 + 2.56i)10-s + (3.28 − 2.06i)11-s + (−2.53 + 4.14i)12-s + (4.92 + 1.12i)13-s + (−3.76 − 0.834i)14-s + (−1.45 + 4.15i)15-s + (−0.991 − 3.87i)16-s + (−1.25 + 1.25i)17-s + ⋯ |
| L(s) = 1 | + (−0.898 + 0.439i)2-s + (−1.39 + 0.157i)3-s + (0.613 − 0.789i)4-s + (0.351 − 0.730i)5-s + (1.18 − 0.754i)6-s + (0.806 + 0.643i)7-s + (−0.203 + 0.979i)8-s + (0.945 − 0.215i)9-s + (0.00526 + 0.810i)10-s + (0.989 − 0.621i)11-s + (−0.731 + 1.19i)12-s + (1.36 + 0.311i)13-s + (−1.00 − 0.223i)14-s + (−0.375 + 1.07i)15-s + (−0.247 − 0.968i)16-s + (−0.304 + 0.304i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.564681 + 0.0483470i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.564681 + 0.0483470i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.27 - 0.621i)T \) |
| 29 | \( 1 + (5.20 + 1.38i)T \) |
| good | 3 | \( 1 + (2.41 - 0.272i)T + (2.92 - 0.667i)T^{2} \) |
| 5 | \( 1 + (-0.786 + 1.63i)T + (-3.11 - 3.90i)T^{2} \) |
| 7 | \( 1 + (-2.13 - 1.70i)T + (1.55 + 6.82i)T^{2} \) |
| 11 | \( 1 + (-3.28 + 2.06i)T + (4.77 - 9.91i)T^{2} \) |
| 13 | \( 1 + (-4.92 - 1.12i)T + (11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (1.25 - 1.25i)T - 17iT^{2} \) |
| 19 | \( 1 + (-0.576 + 5.11i)T + (-18.5 - 4.22i)T^{2} \) |
| 23 | \( 1 + (2.02 + 4.20i)T + (-14.3 + 17.9i)T^{2} \) |
| 31 | \( 1 + (-3.32 - 9.50i)T + (-24.2 + 19.3i)T^{2} \) |
| 37 | \( 1 + (-0.203 + 0.324i)T + (-16.0 - 33.3i)T^{2} \) |
| 41 | \( 1 + (-0.984 - 0.984i)T + 41iT^{2} \) |
| 43 | \( 1 + (5.31 + 1.85i)T + (33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (-1.43 - 2.28i)T + (-20.3 + 42.3i)T^{2} \) |
| 53 | \( 1 + (1.24 + 0.599i)T + (33.0 + 41.4i)T^{2} \) |
| 59 | \( 1 - 13.2iT - 59T^{2} \) |
| 61 | \( 1 + (0.350 + 3.11i)T + (-59.4 + 13.5i)T^{2} \) |
| 67 | \( 1 + (0.240 + 1.05i)T + (-60.3 + 29.0i)T^{2} \) |
| 71 | \( 1 + (-0.917 + 4.02i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (11.2 + 3.94i)T + (57.0 + 45.5i)T^{2} \) |
| 79 | \( 1 + (-5.34 + 8.50i)T + (-34.2 - 71.1i)T^{2} \) |
| 83 | \( 1 + (4.36 - 3.47i)T + (18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-9.91 + 3.46i)T + (69.5 - 55.4i)T^{2} \) |
| 97 | \( 1 + (1.65 - 14.7i)T + (-94.5 - 21.5i)T^{2} \) |
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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
−13.61784462713730207641830684796, −12.05937863234635639079949766620, −11.33045161924665227442527285300, −10.65236634997955030310710286328, −9.042632319570500450028293994080, −8.553056400219783824564614087627, −6.64137111785481160471470314236, −5.84468519365887574850350933328, −4.82931164298639013388147094175, −1.29255936899897533074873111527,
1.46326979090917218361666824227, 3.96343401837745859580637279450, 5.94964914605288681298257190518, 6.86085350353718117684002460308, 8.021674352351265353737053009086, 9.645559166035400489605439648549, 10.63177020660769797395725610110, 11.29945746110335524624639250232, 11.93590819674168381752803672254, 13.26036346595945883734699727789
