| L(s) = 1 | + (0.673 − 1.24i)2-s + (0.988 + 2.38i)3-s + (−1.09 − 1.67i)4-s + (−0.453 − 0.891i)5-s + (3.63 + 0.379i)6-s + (0.728 − 3.03i)7-s + (−2.81 + 0.227i)8-s + (−2.59 + 2.59i)9-s + (−1.41 − 0.0360i)10-s + (−3.15 − 3.69i)11-s + (2.92 − 4.26i)12-s + (1.13 − 1.85i)13-s + (−3.28 − 2.94i)14-s + (1.67 − 1.96i)15-s + (−1.61 + 3.65i)16-s + (0.376 − 4.77i)17-s + ⋯ |
| L(s) = 1 | + (0.476 − 0.879i)2-s + (0.570 + 1.37i)3-s + (−0.545 − 0.837i)4-s + (−0.203 − 0.398i)5-s + (1.48 + 0.154i)6-s + (0.275 − 1.14i)7-s + (−0.996 + 0.0805i)8-s + (−0.866 + 0.866i)9-s + (−0.447 − 0.0113i)10-s + (−0.952 − 1.11i)11-s + (0.843 − 1.23i)12-s + (0.315 − 0.514i)13-s + (−0.876 − 0.788i)14-s + (0.433 − 0.507i)15-s + (−0.404 + 0.914i)16-s + (0.0912 − 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.353 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.01707 - 1.47174i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01707 - 1.47174i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.673 + 1.24i)T \) |
| 5 | \( 1 + (0.453 + 0.891i)T \) |
| 41 | \( 1 + (-6.27 + 1.27i)T \) |
| good | 3 | \( 1 + (-0.988 - 2.38i)T + (-2.12 + 2.12i)T^{2} \) |
| 7 | \( 1 + (-0.728 + 3.03i)T + (-6.23 - 3.17i)T^{2} \) |
| 11 | \( 1 + (3.15 + 3.69i)T + (-1.72 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.13 + 1.85i)T + (-5.90 - 11.5i)T^{2} \) |
| 17 | \( 1 + (-0.376 + 4.77i)T + (-16.7 - 2.65i)T^{2} \) |
| 19 | \( 1 + (4.81 - 2.95i)T + (8.62 - 16.9i)T^{2} \) |
| 23 | \( 1 + (-3.76 + 2.73i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (0.409 + 5.20i)T + (-28.6 + 4.53i)T^{2} \) |
| 31 | \( 1 + (-1.83 - 5.64i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.22 + 3.76i)T + (-29.9 - 21.7i)T^{2} \) |
| 43 | \( 1 + (0.841 - 5.31i)T + (-40.8 - 13.2i)T^{2} \) |
| 47 | \( 1 + (-2.10 - 8.77i)T + (-41.8 + 21.3i)T^{2} \) |
| 53 | \( 1 + (-5.33 + 0.419i)T + (52.3 - 8.29i)T^{2} \) |
| 59 | \( 1 + (4.07 + 5.60i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (1.06 + 6.74i)T + (-58.0 + 18.8i)T^{2} \) |
| 67 | \( 1 + (10.0 + 8.60i)T + (10.4 + 66.1i)T^{2} \) |
| 71 | \( 1 + (3.05 - 2.61i)T + (11.1 - 70.1i)T^{2} \) |
| 73 | \( 1 + (-3.18 - 3.18i)T + 73iT^{2} \) |
| 79 | \( 1 + (-3.40 + 1.40i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + 2.42iT - 83T^{2} \) |
| 89 | \( 1 + (-14.5 - 3.49i)T + (79.2 + 40.4i)T^{2} \) |
| 97 | \( 1 + (8.25 + 7.05i)T + (15.1 + 95.8i)T^{2} \) |
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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
−10.22659388231620499212595416746, −9.329039405284993314374550776919, −8.563716083437831140726639075615, −7.76056693560414619068577741594, −6.08019270895680473068346733030, −4.98142093416965330042458288976, −4.41140220162833968989042458366, −3.53101926379721723405822087156, −2.73662509742462032514139443596, −0.70166495999243350727720925299,
2.03211849737343881672222443245, 2.80564838054611830887655389503, 4.26678635630321247579161646920, 5.43780840819567593538862512816, 6.34843863803204169845847331362, 7.09992350802380145957165217948, 7.74366669736844232399378903311, 8.549203472459751475303769844195, 9.071169407967495417395591153107, 10.52860222435439421091345666458
