| L(s) = 1 | + (−1.90 − 0.619i)2-s + (−1.63 + 2.51i)3-s + (0.0135 + 0.00987i)4-s + (5.21 − 1.69i)5-s + (4.67 − 3.78i)6-s + (4.52 + 3.28i)7-s + (4.69 + 6.45i)8-s + (−3.66 − 8.22i)9-s − 10.9·10-s + (−0.0470 + 0.0180i)12-s + (3.00 − 9.24i)13-s + (−6.58 − 9.06i)14-s + (−4.25 + 15.8i)15-s + (−4.96 − 15.2i)16-s + (−16.9 + 5.52i)17-s + (1.88 + 17.9i)18-s + ⋯ |
| L(s) = 1 | + (−0.953 − 0.309i)2-s + (−0.544 + 0.838i)3-s + (0.00339 + 0.00246i)4-s + (1.04 − 0.338i)5-s + (0.778 − 0.630i)6-s + (0.646 + 0.469i)7-s + (0.586 + 0.807i)8-s + (−0.406 − 0.913i)9-s − 1.09·10-s + (−0.00392 + 0.00150i)12-s + (0.230 − 0.710i)13-s + (−0.470 − 0.647i)14-s + (−0.283 + 1.05i)15-s + (−0.310 − 0.955i)16-s + (−0.999 + 0.324i)17-s + (0.104 + 0.996i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0304i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.982205 + 0.0149762i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.982205 + 0.0149762i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.63 - 2.51i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (1.90 + 0.619i)T + (3.23 + 2.35i)T^{2} \) |
| 5 | \( 1 + (-5.21 + 1.69i)T + (20.2 - 14.6i)T^{2} \) |
| 7 | \( 1 + (-4.52 - 3.28i)T + (15.1 + 46.6i)T^{2} \) |
| 13 | \( 1 + (-3.00 + 9.24i)T + (-136. - 99.3i)T^{2} \) |
| 17 | \( 1 + (16.9 - 5.52i)T + (233. - 169. i)T^{2} \) |
| 19 | \( 1 + (-15.0 + 10.9i)T + (111. - 343. i)T^{2} \) |
| 23 | \( 1 - 12.3iT - 529T^{2} \) |
| 29 | \( 1 + (-1.45 + 2.00i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (-15.2 + 46.8i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (-31.8 - 23.1i)T + (423. + 1.30e3i)T^{2} \) |
| 41 | \( 1 + (-33.2 - 45.7i)T + (-519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 - 43.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-33.9 - 46.7i)T + (-682. + 2.10e3i)T^{2} \) |
| 53 | \( 1 + (-41.0 - 13.3i)T + (2.27e3 + 1.65e3i)T^{2} \) |
| 59 | \( 1 + (-52.9 + 72.8i)T + (-1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-9.53 - 29.3i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 - 34.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + (35.7 - 11.6i)T + (4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (9.81 + 7.12i)T + (1.64e3 + 5.06e3i)T^{2} \) |
| 79 | \( 1 + (-19.5 + 60.0i)T + (-5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-9.22 + 2.99i)T + (5.57e3 - 4.04e3i)T^{2} \) |
| 89 | \( 1 - 34.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (11.6 - 35.9i)T + (-7.61e3 - 5.53e3i)T^{2} \) |
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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
−11.12055579866259263330796933100, −10.07874791101229517155987710332, −9.488177389160875133501395865074, −8.852461394145944560281261133181, −7.84233240989814865793545273152, −6.07314911827836885641359861030, −5.36583051656213123129272993945, −4.44322988097124143311471351307, −2.44298193785771087230921645348, −0.945066903836358747274923995209,
0.981037062714141615566084749747, 2.19164304065141788144265792194, 4.34327821606802223237516608735, 5.66165271122500982643825732518, 6.75214545395280338046239889062, 7.32815585627535081573831594727, 8.399121811422722891442548094816, 9.238365080269735221767713821629, 10.32503863642941878089631352036, 10.93212863567108823254413941048
