L(s) = 1 | + 8.11i·5-s − 9.48i·7-s + 10.5·11-s + 20.9i·13-s + 4.92·17-s + 26.9·19-s − 43.7i·23-s − 40.9·25-s − 14.5i·29-s + 25.4i·31-s + 77.0·35-s + 12.9i·37-s + 50.1·41-s + 5.03·43-s − 40.8i·47-s + ⋯ |
L(s) = 1 | + 1.62i·5-s − 1.35i·7-s + 0.962·11-s + 1.61i·13-s + 0.289·17-s + 1.41·19-s − 1.90i·23-s − 1.63·25-s − 0.500i·29-s + 0.822i·31-s + 2.20·35-s + 0.350i·37-s + 1.22·41-s + 0.117·43-s − 0.869i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.356676824\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.356676824\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 8.11iT - 25T^{2} \) |
| 7 | \( 1 + 9.48iT - 49T^{2} \) |
| 11 | \( 1 - 10.5T + 121T^{2} \) |
| 13 | \( 1 - 20.9iT - 169T^{2} \) |
| 17 | \( 1 - 4.92T + 289T^{2} \) |
| 19 | \( 1 - 26.9T + 361T^{2} \) |
| 23 | \( 1 + 43.7iT - 529T^{2} \) |
| 29 | \( 1 + 14.5iT - 841T^{2} \) |
| 31 | \( 1 - 25.4iT - 961T^{2} \) |
| 37 | \( 1 - 12.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 50.1T + 1.68e3T^{2} \) |
| 43 | \( 1 - 5.03T + 1.84e3T^{2} \) |
| 47 | \( 1 + 40.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 27.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 24.0T + 3.48e3T^{2} \) |
| 61 | \( 1 + 28.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 65.7T + 4.48e3T^{2} \) |
| 71 | \( 1 + 90.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 127.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 35.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 101.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 15.6T + 7.92e3T^{2} \) |
| 97 | \( 1 - 73.8T + 9.40e3T^{2} \) |
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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
−9.112197899699554208170572589718, −7.892460530540782950500524679345, −7.17288549291839372065034801803, −6.69678165968561395907850199482, −6.22284491238830336891307981368, −4.69689045187688367300016143065, −3.94543376102344379026990128210, −3.26674404620747507996881138663, −2.16008208256046431292511224820, −0.915520291802682587465827478807,
0.77717641527723026535037417152, 1.57684433859005953747986112550, 2.92127205934222467628356453121, 3.83982628161574870538438774110, 5.02792481634043405969942153864, 5.54321220349094601137358620784, 5.95597530871976674818010064951, 7.51202770902770713550279157878, 7.998989870834730320951402963821, 8.887847475571140908465244834309