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
−13.37361357947308031526903151455, −12.30060797024844820549191346693, −10.68424076775610041035040225197, −9.997745251699011549536580558818, −8.858827311475035023000321179177, −7.47957402290085204008792120805, −6.87633727866159150076650645558, −4.73555273405770835561729987144, −3.24061966802244000571514721168, −0.093217854039384950487022183487,
2.44366532663247253954105765378, 4.75599055275664903326882220842, 5.87341168133823870480966397250, 7.73524698390887114560272893541, 8.806220057587467058113710850275, 9.696313724247921195135820630090, 10.53707943012629042912659356608, 12.13635817728274518380874810829, 12.78773399422022103806552887373, 14.04491783747886284685692662202