| L(s) = 1 | − 5.19·3-s − 23.7i·5-s + 9.38i·7-s + 27·9-s − 112.·11-s − 56.5i·13-s + 123. i·15-s − 79.9·17-s − 211.·19-s − 48.7i·21-s − 217. i·23-s + 60.9·25-s − 140.·27-s − 616. i·29-s + 1.11e3i·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.949i·5-s + 0.191i·7-s + 0.333·9-s − 0.925·11-s − 0.334i·13-s + 0.548i·15-s − 0.276·17-s − 0.585·19-s − 0.110i·21-s − 0.410i·23-s + 0.0975·25-s − 0.192·27-s − 0.733i·29-s + 1.15i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.5892062737\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5892062737\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 5.19T \) |
| good | 5 | \( 1 + 23.7iT - 625T^{2} \) |
| 7 | \( 1 - 9.38iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 112.T + 1.46e4T^{2} \) |
| 13 | \( 1 + 56.5iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 79.9T + 8.35e4T^{2} \) |
| 19 | \( 1 + 211.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 217. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 616. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.11e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 802. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 2.41e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 2.13e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 3.59e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 833. iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 1.30e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 4.78e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 4.02e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 9.48e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 266.T + 2.83e7T^{2} \) |
| 79 | \( 1 - 5.75e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 7.28e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 1.41e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 3.11e3T + 8.85e7T^{2} \) |
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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
−10.86314475389270167761466565729, −10.19010647311904995891036786010, −9.006551374029201094455536808072, −8.278144199521341093730184100306, −7.18177615093271996095265809414, −5.95378923784809770029506212116, −5.13457804988755334532501086257, −4.24973012910214436680436429008, −2.56927416791414432896579464985, −1.03795742359696888292648267858,
0.20660057759175038648122353339, 2.03506882806073001920297222615, 3.30773212524639191830132547631, 4.60370023381770214802384512248, 5.72373033154459786337337160309, 6.72590594649924858000178276802, 7.43769538110321253308535876817, 8.593910399595611133312960479205, 9.867853606430114656853453110181, 10.60803191637864781029460406078
