| L(s) = 1 | + (−3.91 + 3.91i)3-s + (−4.72 + 4.72i)5-s + 45.3·7-s + 50.3i·9-s + (−110. − 110. i)11-s + (157. + 157. i)13-s − 36.9i·15-s − 378.·17-s + (−203. + 203. i)19-s + (−177. + 177. i)21-s − 740.·23-s + 580. i·25-s + (−514. − 514. i)27-s + (−82.6 − 82.6i)29-s − 286. i·31-s + ⋯ |
| L(s) = 1 | + (−0.434 + 0.434i)3-s + (−0.188 + 0.188i)5-s + 0.925·7-s + 0.621i·9-s + (−0.910 − 0.910i)11-s + (0.929 + 0.929i)13-s − 0.164i·15-s − 1.31·17-s + (−0.562 + 0.562i)19-s + (−0.402 + 0.402i)21-s − 1.39·23-s + 0.928i·25-s + (−0.705 − 0.705i)27-s + (−0.0982 − 0.0982i)29-s − 0.297i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 - 0.457i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.889 - 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.182937 + 0.755361i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.182937 + 0.755361i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + (3.91 - 3.91i)T - 81iT^{2} \) |
| 5 | \( 1 + (4.72 - 4.72i)T - 625iT^{2} \) |
| 7 | \( 1 - 45.3T + 2.40e3T^{2} \) |
| 11 | \( 1 + (110. + 110. i)T + 1.46e4iT^{2} \) |
| 13 | \( 1 + (-157. - 157. i)T + 2.85e4iT^{2} \) |
| 17 | \( 1 + 378.T + 8.35e4T^{2} \) |
| 19 | \( 1 + (203. - 203. i)T - 1.30e5iT^{2} \) |
| 23 | \( 1 + 740.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (82.6 + 82.6i)T + 7.07e5iT^{2} \) |
| 31 | \( 1 + 286. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + (1.47e3 - 1.47e3i)T - 1.87e6iT^{2} \) |
| 41 | \( 1 - 1.30e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + (-366. - 366. i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + 751. iT - 4.87e6T^{2} \) |
| 53 | \( 1 + (-1.92e3 + 1.92e3i)T - 7.89e6iT^{2} \) |
| 59 | \( 1 + (-1.35e3 - 1.35e3i)T + 1.21e7iT^{2} \) |
| 61 | \( 1 + (-1.83e3 - 1.83e3i)T + 1.38e7iT^{2} \) |
| 67 | \( 1 + (2.20e3 - 2.20e3i)T - 2.01e7iT^{2} \) |
| 71 | \( 1 - 8.97e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 9.35e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 2.86e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (-1.03e3 + 1.03e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 - 5.17e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 8.53e3T + 8.85e7T^{2} \) |
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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
−13.29981118013829432079206445953, −11.59348760461196207022916201129, −11.11904138979044991149374134566, −10.27415043199377014098373866324, −8.664607335048198235645718566363, −7.904951109907914974026352468307, −6.31181203454675091429182990711, −5.12177978617229329500345981411, −3.97229842411239499198710546147, −1.97710864263946820925695107590,
0.32788570262435370115277626504, 2.07143225294614463861629203057, 4.14123043348694948176047826125, 5.42220392615169268901238165214, 6.67616786795269967807145903143, 7.88271377060738214888572472096, 8.815366382738124865336619004861, 10.37035335503331641881317437444, 11.19725301906008736275098353580, 12.30554304586642228232433622589
