L(s) = 1 | − 2.73i·5-s − 7-s − 5.46i·11-s − 6.73i·13-s − 2·17-s − 1.26i·19-s − 3.46·23-s − 2.46·25-s − 1.46i·29-s − 4·31-s + 2.73i·35-s − 1.46i·37-s + 2·41-s + 5.46i·43-s + 2.92·47-s + ⋯ |
L(s) = 1 | − 1.22i·5-s − 0.377·7-s − 1.64i·11-s − 1.86i·13-s − 0.485·17-s − 0.290i·19-s − 0.722·23-s − 0.492·25-s − 0.271i·29-s − 0.718·31-s + 0.461i·35-s − 0.240i·37-s + 0.312·41-s + 0.833i·43-s + 0.427·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.091175556\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.091175556\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + 2.73iT - 5T^{2} \) |
| 11 | \( 1 + 5.46iT - 11T^{2} \) |
| 13 | \( 1 + 6.73iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 1.26iT - 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 + 1.46iT - 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 1.46iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 5.46iT - 43T^{2} \) |
| 47 | \( 1 - 2.92T + 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 - 9.66iT - 59T^{2} \) |
| 61 | \( 1 - 11.1iT - 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 2.92T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 5.66iT - 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 - 8.92T + 97T^{2} \) |
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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
−8.294223623181377900101168035785, −7.51146070424681157701999803687, −6.39138192700564970104563657955, −5.62037114378331800433256324955, −5.30950037888657056992897584542, −4.19023035457148739337713994622, −3.39418598480175409499983283713, −2.54175496718756926735722025659, −1.03903121379069734055922913777, −0.34492076556418182734895864822,
1.87035801753806654882220904304, 2.29918942516118968933742609884, 3.52535459676349387984507876790, 4.20620152303346507327336569959, 4.98155777499815363662452736756, 6.22346244067086025248847766358, 6.71746048674274947607776783369, 7.17850952055671448721370334649, 7.892389565834149131938256886691, 9.113420112540729419049667009918