L(s) = 1 | + (−0.433 − 0.900i)2-s + (−0.623 + 0.781i)4-s + (0.974 + 0.222i)8-s + (0.826 + 0.563i)11-s + (0.563 − 0.826i)13-s + (−0.222 − 0.974i)16-s + (−0.149 − 0.988i)17-s + (0.5 + 0.866i)19-s + (0.149 − 0.988i)22-s + (0.930 − 0.365i)23-s + (−0.988 − 0.149i)26-s + (0.988 − 0.149i)29-s + 31-s + (−0.781 + 0.623i)32-s + (−0.826 + 0.563i)34-s + ⋯ |
L(s) = 1 | + (−0.433 − 0.900i)2-s + (−0.623 + 0.781i)4-s + (0.974 + 0.222i)8-s + (0.826 + 0.563i)11-s + (0.563 − 0.826i)13-s + (−0.222 − 0.974i)16-s + (−0.149 − 0.988i)17-s + (0.5 + 0.866i)19-s + (0.149 − 0.988i)22-s + (0.930 − 0.365i)23-s + (−0.988 − 0.149i)26-s + (0.988 − 0.149i)29-s + 31-s + (−0.781 + 0.623i)32-s + (−0.826 + 0.563i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.171 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.171 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.722996193 - 1.448823904i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.722996193 - 1.448823904i\) |
\(L(1)\) |
\(\approx\) |
\(0.9409548281 - 0.4231877493i\) |
\(L(1)\) |
\(\approx\) |
\(0.9409548281 - 0.4231877493i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.433 - 0.900i)T \) |
| 11 | \( 1 + (0.826 + 0.563i)T \) |
| 13 | \( 1 + (0.563 - 0.826i)T \) |
| 17 | \( 1 + (-0.149 - 0.988i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.930 - 0.365i)T \) |
| 29 | \( 1 + (0.988 - 0.149i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (0.930 + 0.365i)T \) |
| 41 | \( 1 + (0.955 - 0.294i)T \) |
| 43 | \( 1 + (0.294 - 0.955i)T \) |
| 47 | \( 1 + (-0.433 - 0.900i)T \) |
| 53 | \( 1 + (0.930 - 0.365i)T \) |
| 59 | \( 1 + (0.222 + 0.974i)T \) |
| 61 | \( 1 + (0.623 + 0.781i)T \) |
| 67 | \( 1 + iT \) |
| 71 | \( 1 + (0.623 - 0.781i)T \) |
| 73 | \( 1 + (-0.563 - 0.826i)T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + (-0.563 - 0.826i)T \) |
| 89 | \( 1 + (-0.0747 - 0.997i)T \) |
| 97 | \( 1 + (-0.866 - 0.5i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.31574237510708712619322229741, −19.16007614813677897992257166958, −18.079419412365924023174555501665, −17.479614101522634006711790523868, −16.84618304228103154263410332956, −16.11310583575846775448534373401, −15.548071625555771710658967394012, −14.658143485922311144220066113465, −14.08541266796328015467064393723, −13.409181869351648492891855900216, −12.600466411862328679539894110490, −11.307538593656293109576062097780, −11.04822180078659153633234021787, −9.83365344922486664855558129637, −9.23557052038980057846419900815, −8.5689307736440237235281604896, −7.89086183268267167740681350144, −6.78761533348184054400081163543, −6.43763279383523128813033503652, −5.59707259071727141274180267973, −4.57174812546651188748643694359, −3.96410963637153378444827009053, −2.763475278559118999531049605639, −1.37844526639336158831410337967, −0.84175783920184471313048159763,
0.670112595224903869344158617027, 1.20891094371505180868744754033, 2.403917455794236808116365410498, 3.098165543058767905157507779401, 4.01685144877193253646466430352, 4.74333707070896949677232261747, 5.72691975815341239232450048056, 6.85598384122653134068150059529, 7.576502494584272436564921628340, 8.49148076852716575649732021071, 9.06218388307559993634003938574, 10.00673132779575687814158780957, 10.38392201130562129680188665906, 11.50352276783027481737084970146, 11.87167122515096894729620143935, 12.709138915717812661902453238421, 13.43655390447283634538666803656, 14.13882199471409335612775156068, 14.99160471802397199291936245343, 15.96906161292451200781551625481, 16.65706955153997629179940977082, 17.48021700492791250875306023139, 18.01808131589161393955245694970, 18.68170867613989690681127018179, 19.47888403453884530592606637653