| L(s) = 1 | + 5.19·3-s + 15.5·5-s − 93.8i·7-s + 27·9-s + (−60.9 + 104. i)11-s − 29.4i·13-s + 80.8·15-s − 251. i·17-s − 80.2i·19-s − 487. i·21-s + 702.·23-s − 383.·25-s + 140.·27-s − 1.44e3i·29-s − 1.27e3·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.622·5-s − 1.91i·7-s + 0.333·9-s + (−0.503 + 0.863i)11-s − 0.174i·13-s + 0.359·15-s − 0.871i·17-s − 0.222i·19-s − 1.10i·21-s + 1.32·23-s − 0.612·25-s + 0.192·27-s − 1.72i·29-s − 1.33·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.503 + 0.863i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.503 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.173242350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.173242350\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 5.19T \) |
| 11 | \( 1 + (60.9 - 104. i)T \) |
| good | 5 | \( 1 - 15.5T + 625T^{2} \) |
| 7 | \( 1 + 93.8iT - 2.40e3T^{2} \) |
| 13 | \( 1 + 29.4iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 251. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 80.2iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 702.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 1.44e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.27e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 115.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 1.07e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 1.88e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 1.59e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 1.19e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 1.89e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 3.77e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 1.25e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 3.59e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 1.75e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 6.74e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 8.61e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 9.33e3T + 6.27e7T^{2} \) |
| 97 | \( 1 - 1.10e4T + 8.85e7T^{2} \) |
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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
−9.845661306325519539454465942239, −9.380548076731349954190254921571, −7.87374952779918057647191558534, −7.39622800892541805698357512230, −6.51717922337008628084367410972, −5.02681091611134623540748050436, −4.19937710000508079142651973448, −3.03054694536655040409033361901, −1.74600445000380665995600877249, −0.47929545171875344011469292576,
1.64132692531162613739628390481, 2.56114477933822682782015265531, 3.51405911170075611558784027875, 5.28787397960501395312191581488, 5.69845348540007289019420486705, 6.84747848387616889034382487141, 8.189758381729369151550073582213, 8.854689532468491399250075671081, 9.337414094761026199834569444085, 10.51745040224734439249960187222
