| L(s) = 1 | + (0.0886 + 0.121i)2-s + (4.93 − 15.1i)4-s + (−11.7 − 8.54i)5-s + (66.7 + 21.6i)7-s + (4.58 − 1.49i)8-s − 2.19i·10-s + (−116. − 33.5i)11-s + (−146. − 201. i)13-s + (3.26 + 10.0i)14-s + (−206. − 149. i)16-s + (201. − 276. i)17-s + (−66.2 + 21.5i)19-s + (−187. + 136. i)20-s + (−6.21 − 17.1i)22-s + 789.·23-s + ⋯ |
| L(s) = 1 | + (0.0221 + 0.0304i)2-s + (0.308 − 0.949i)4-s + (−0.470 − 0.341i)5-s + (1.36 + 0.442i)7-s + (0.0716 − 0.0232i)8-s − 0.0219i·10-s + (−0.960 − 0.276i)11-s + (−0.865 − 1.19i)13-s + (0.0166 + 0.0513i)14-s + (−0.805 − 0.585i)16-s + (0.695 − 0.957i)17-s + (−0.183 + 0.0596i)19-s + (−0.469 + 0.341i)20-s + (−0.0128 − 0.0354i)22-s + 1.49·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.209 + 0.977i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.209 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.999639 - 1.23650i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.999639 - 1.23650i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (116. + 33.5i)T \) |
| good | 2 | \( 1 + (-0.0886 - 0.121i)T + (-4.94 + 15.2i)T^{2} \) |
| 5 | \( 1 + (11.7 + 8.54i)T + (193. + 594. i)T^{2} \) |
| 7 | \( 1 + (-66.7 - 21.6i)T + (1.94e3 + 1.41e3i)T^{2} \) |
| 13 | \( 1 + (146. + 201. i)T + (-8.82e3 + 2.71e4i)T^{2} \) |
| 17 | \( 1 + (-201. + 276. i)T + (-2.58e4 - 7.94e4i)T^{2} \) |
| 19 | \( 1 + (66.2 - 21.5i)T + (1.05e5 - 7.66e4i)T^{2} \) |
| 23 | \( 1 - 789.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (1.01e3 + 329. i)T + (5.72e5 + 4.15e5i)T^{2} \) |
| 31 | \( 1 + (-44.9 + 32.6i)T + (2.85e5 - 8.78e5i)T^{2} \) |
| 37 | \( 1 + (200. - 617. i)T + (-1.51e6 - 1.10e6i)T^{2} \) |
| 41 | \( 1 + (49.1 - 15.9i)T + (2.28e6 - 1.66e6i)T^{2} \) |
| 43 | \( 1 - 923. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (-890. - 2.73e3i)T + (-3.94e6 + 2.86e6i)T^{2} \) |
| 53 | \( 1 + (-4.29e3 + 3.12e3i)T + (2.43e6 - 7.50e6i)T^{2} \) |
| 59 | \( 1 + (595. - 1.83e3i)T + (-9.80e6 - 7.12e6i)T^{2} \) |
| 61 | \( 1 + (-1.82e3 + 2.50e3i)T + (-4.27e6 - 1.31e7i)T^{2} \) |
| 67 | \( 1 + 105.T + 2.01e7T^{2} \) |
| 71 | \( 1 + (-4.31e3 - 3.13e3i)T + (7.85e6 + 2.41e7i)T^{2} \) |
| 73 | \( 1 + (-6.02e3 - 1.95e3i)T + (2.29e7 + 1.66e7i)T^{2} \) |
| 79 | \( 1 + (-4.30e3 - 5.91e3i)T + (-1.20e7 + 3.70e7i)T^{2} \) |
| 83 | \( 1 + (4.17e3 - 5.73e3i)T + (-1.46e7 - 4.51e7i)T^{2} \) |
| 89 | \( 1 + 515.T + 6.27e7T^{2} \) |
| 97 | \( 1 + (-7.46e3 + 5.42e3i)T + (2.73e7 - 8.41e7i)T^{2} \) |
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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
−12.81766471231694116457352126548, −11.62769310374425341491225201287, −10.85704731684243716766787408762, −9.735290654165452987446498827195, −8.279629580801512764908317583885, −7.40204244647268746482310424525, −5.46241350619981425709900370539, −4.95184399101578896200834770558, −2.52923841020484346676950356103, −0.73143339838500982400911372653,
2.05961406828494106922264399856, 3.77064287952146351545401892583, 5.02304137523730285091475651285, 7.19359343045863024441184343266, 7.64740904659231290390033678074, 8.852635353016668035896174664681, 10.61315566305783611164146945584, 11.36978491736604319942915631838, 12.28703566320357743774044632672, 13.37969383010009946386677847074
