| L(s) = 1 | + (−1.28 − 0.419i)2-s + (−1.74 − 1.27i)4-s + (−0.708 − 2.18i)5-s + (−5.74 + 7.90i)7-s + (4.91 + 6.75i)8-s + 3.11i·10-s + (−10.3 + 3.76i)11-s + (−14.6 − 4.75i)13-s + (10.7 − 7.78i)14-s + (−0.830 − 2.55i)16-s + (−11.2 + 3.65i)17-s + (7.10 + 9.77i)19-s + (−1.53 + 4.71i)20-s + (14.9 − 0.531i)22-s − 16.6·23-s + ⋯ |
| L(s) = 1 | + (−0.644 − 0.209i)2-s + (−0.437 − 0.317i)4-s + (−0.141 − 0.436i)5-s + (−0.820 + 1.12i)7-s + (0.613 + 0.844i)8-s + 0.311i·10-s + (−0.939 + 0.342i)11-s + (−1.12 − 0.365i)13-s + (0.765 − 0.556i)14-s + (−0.0519 − 0.159i)16-s + (−0.661 + 0.214i)17-s + (0.373 + 0.514i)19-s + (−0.0765 + 0.235i)20-s + (0.677 − 0.0241i)22-s − 0.724·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.576i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0313175 + 0.0986684i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0313175 + 0.0986684i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (10.3 - 3.76i)T \) |
| good | 2 | \( 1 + (1.28 + 0.419i)T + (3.23 + 2.35i)T^{2} \) |
| 5 | \( 1 + (0.708 + 2.18i)T + (-20.2 + 14.6i)T^{2} \) |
| 7 | \( 1 + (5.74 - 7.90i)T + (-15.1 - 46.6i)T^{2} \) |
| 13 | \( 1 + (14.6 + 4.75i)T + (136. + 99.3i)T^{2} \) |
| 17 | \( 1 + (11.2 - 3.65i)T + (233. - 169. i)T^{2} \) |
| 19 | \( 1 + (-7.10 - 9.77i)T + (-111. + 343. i)T^{2} \) |
| 23 | \( 1 + 16.6T + 529T^{2} \) |
| 29 | \( 1 + (-15.6 + 21.4i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (-1.28 + 3.95i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (54.4 + 39.5i)T + (423. + 1.30e3i)T^{2} \) |
| 41 | \( 1 + (-10.6 - 14.6i)T + (-519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 - 46.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-49.2 + 35.8i)T + (682. - 2.10e3i)T^{2} \) |
| 53 | \( 1 + (31.2 - 96.2i)T + (-2.27e3 - 1.65e3i)T^{2} \) |
| 59 | \( 1 + (78.7 + 57.1i)T + (1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-1.19 + 0.388i)T + (3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + 55.1T + 4.48e3T^{2} \) |
| 71 | \( 1 + (-4.62 - 14.2i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (44.6 - 61.3i)T + (-1.64e3 - 5.06e3i)T^{2} \) |
| 79 | \( 1 + (-27.7 - 9.02i)T + (5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-6.06 + 1.96i)T + (5.57e3 - 4.04e3i)T^{2} \) |
| 89 | \( 1 - 3.95T + 7.92e3T^{2} \) |
| 97 | \( 1 + (10.2 - 31.5i)T + (-7.61e3 - 5.53e3i)T^{2} \) |
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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
−14.04491783747886284685692662202, −12.78773399422022103806552887373, −12.13635817728274518380874810829, −10.53707943012629042912659356608, −9.696313724247921195135820630090, −8.806220057587467058113710850275, −7.73524698390887114560272893541, −5.87341168133823870480966397250, −4.75599055275664903326882220842, −2.44366532663247253954105765378,
0.093217854039384950487022183487, 3.24061966802244000571514721168, 4.73555273405770835561729987144, 6.87633727866159150076650645558, 7.47957402290085204008792120805, 8.858827311475035023000321179177, 9.997745251699011549536580558818, 10.68424076775610041035040225197, 12.30060797024844820549191346693, 13.37361357947308031526903151455
