L(s) = 1 | + 1.73i·3-s − i·5-s + 7-s + i·11-s − i·13-s + 1.73·15-s − 2.26·17-s + 2i·19-s + 1.73i·21-s + 3.46·23-s − 25-s + 5.19i·27-s + 1.19i·29-s + 4·31-s − 1.73·33-s + ⋯ |
L(s) = 1 | + 0.999i·3-s − 0.447i·5-s + 0.377·7-s + 0.301i·11-s − 0.277i·13-s + 0.447·15-s − 0.550·17-s + 0.458i·19-s + 0.377i·21-s + 0.722·23-s − 0.200·25-s + 1.00i·27-s + 0.222i·29-s + 0.718·31-s − 0.301·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.874208897\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.874208897\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 - 1.73iT - 3T^{2} \) |
| 11 | \( 1 - iT - 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 + 2.26T + 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 - 1.19iT - 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 0.535iT - 37T^{2} \) |
| 41 | \( 1 - 2.92T + 41T^{2} \) |
| 43 | \( 1 - 3.46iT - 43T^{2} \) |
| 47 | \( 1 - 9.92T + 47T^{2} \) |
| 53 | \( 1 - 1.46iT - 53T^{2} \) |
| 59 | \( 1 - 9.46iT - 59T^{2} \) |
| 61 | \( 1 - 1.46iT - 61T^{2} \) |
| 67 | \( 1 + 5.46iT - 67T^{2} \) |
| 71 | \( 1 - 7.46T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 2.26T + 79T^{2} \) |
| 83 | \( 1 - 8.53iT - 83T^{2} \) |
| 89 | \( 1 - 2.53T + 89T^{2} \) |
| 97 | \( 1 + 17.7T + 97T^{2} \) |
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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
−9.247962192032913768527088558016, −8.602613103228379147494702364490, −7.74321499086003371322074344411, −6.92680746449745076261584401882, −5.85688683730551518626843096004, −5.01695755611888114516715702097, −4.43516062559659256863195598025, −3.67682321486881283853588692878, −2.48896253181878358150426487108, −1.15973564745637400311973883715,
0.75709439623621517590616315508, 1.94213566134547477704475188163, 2.76346474051351013311277788204, 3.98099604121805060501299814626, 4.89515310649093579701171413291, 5.95144176814593440917898760768, 6.71668262430531394293961579893, 7.21502695667503979609053011257, 7.997506901534562814021816792426, 8.725128357527132314357312962899