| L(s) = 1 | + (−0.483 + 1.32i)2-s + (−0.570 + 2.94i)3-s + (−1.53 − 1.28i)4-s + (3.98 + 0.702i)5-s + (−3.63 − 2.18i)6-s + (−10.1 + 8.49i)7-s + (2.44 − 1.41i)8-s + (−8.34 − 3.36i)9-s + (−2.86 + 4.95i)10-s + (13.3 − 2.35i)11-s + (4.66 − 3.77i)12-s + (17.4 − 6.34i)13-s + (−6.38 − 17.5i)14-s + (−4.34 + 11.3i)15-s + (0.694 + 3.93i)16-s + (13.8 + 8.01i)17-s + ⋯ |
| L(s) = 1 | + (−0.241 + 0.664i)2-s + (−0.190 + 0.981i)3-s + (−0.383 − 0.321i)4-s + (0.797 + 0.140i)5-s + (−0.606 − 0.363i)6-s + (−1.44 + 1.21i)7-s + (0.306 − 0.176i)8-s + (−0.927 − 0.373i)9-s + (−0.286 + 0.495i)10-s + (1.21 − 0.214i)11-s + (0.388 − 0.314i)12-s + (1.34 − 0.488i)13-s + (−0.456 − 1.25i)14-s + (−0.289 + 0.756i)15-s + (0.0434 + 0.246i)16-s + (0.816 + 0.471i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.506 - 0.862i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.506 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.476736 + 0.833189i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.476736 + 0.833189i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.483 - 1.32i)T \) |
| 3 | \( 1 + (0.570 - 2.94i)T \) |
| good | 5 | \( 1 + (-3.98 - 0.702i)T + (23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (10.1 - 8.49i)T + (8.50 - 48.2i)T^{2} \) |
| 11 | \( 1 + (-13.3 + 2.35i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (-17.4 + 6.34i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-13.8 - 8.01i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-0.327 - 0.566i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-4.24 + 5.05i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (-0.466 + 1.28i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (14.3 + 12.0i)T + (166. + 946. i)T^{2} \) |
| 37 | \( 1 + (8.43 - 14.6i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (14.6 + 40.2i)T + (-1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-0.0113 - 0.0645i)T + (-1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (-30.0 - 35.8i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 - 14.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (4.79 + 0.844i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (12.0 - 10.1i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (37.1 - 13.5i)T + (3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (60.9 + 35.1i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (34.1 + 59.1i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-47.4 - 17.2i)T + (4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-50.2 + 138. i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (17.2 - 9.98i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-18.3 - 104. i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−15.65307975650822432801530542258, −14.69632027819863448680984770470, −13.47788069595621853071062588326, −12.07381275469583082843216979358, −10.44466501175413157190698438740, −9.414558328457004344990495284000, −8.757287012027398137909016332937, −6.20306429713994800271192291206, −5.82612214329808654653613878482, −3.49806634442314215538249746219,
1.25071308830012844554553736169, 3.56463644352525046976960997793, 6.13172776318420528214078018857, 7.15132024219230416899703673744, 9.001375871434019196588235088553, 10.01769694859273845319209684632, 11.34782534792233590028561040616, 12.58110461524586550875371538924, 13.52161848322369218281837179417, 14.01770488140292997348357037420
