L(s) = 1 | + 1.73i·3-s + 5.19i·7-s − 2.99·9-s + 7·13-s + 5.19i·19-s − 9·21-s − 5.19i·27-s − 1.73i·31-s − 10·37-s + 12.1i·39-s − 1.73i·43-s − 20·49-s − 9·57-s − 61-s − 15.5i·63-s + ⋯ |
L(s) = 1 | + 0.999i·3-s + 1.96i·7-s − 0.999·9-s + 1.94·13-s + 1.19i·19-s − 1.96·21-s − 0.999i·27-s − 0.311i·31-s − 1.64·37-s + 1.94i·39-s − 0.264i·43-s − 2.85·49-s − 1.19·57-s − 0.128·61-s − 1.96i·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.483583614\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.483583614\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 5.19iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 7T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 5.19iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 1.73iT - 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 1.73iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 - 12.1iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 17.3iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 19T + 97T^{2} \) |
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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
−10.05769130654097618758111806067, −9.069612726620187540717790281186, −8.701699375239207520995523548335, −8.054530776839527414236421257152, −6.34559182124706036108067872100, −5.80713185390175186171157190897, −5.15650859983305851262442063608, −3.87370064190643972232271268996, −3.12101046085674015410215286326, −1.87657160997623496945792874564,
0.66806757366720855191734551195, 1.57078476498766989933462670894, 3.22351797994124113868420109577, 3.99335382965830650528329180599, 5.19039680580781884217894083783, 6.47864904822841063372522390551, 6.80832493091351651812563599795, 7.70766500544211890785719330700, 8.402956796952966865129499966223, 9.270005683140784836591780933170