L(s) = 1 | + (4.63 + 4.63i)3-s + (29.2 + 29.2i)5-s + 59.6·7-s − 38.0i·9-s + (18.0 − 18.0i)11-s + (−50.7 + 50.7i)13-s + 270. i·15-s − 223.·17-s + (−14.7 − 14.7i)19-s + (276. + 276. i)21-s + 739.·23-s + 1.08e3i·25-s + (551. − 551. i)27-s + (−938. + 938. i)29-s − 938. i·31-s + ⋯ |
L(s) = 1 | + (0.515 + 0.515i)3-s + (1.16 + 1.16i)5-s + 1.21·7-s − 0.469i·9-s + (0.149 − 0.149i)11-s + (−0.300 + 0.300i)13-s + 1.20i·15-s − 0.774·17-s + (−0.0408 − 0.0408i)19-s + (0.626 + 0.626i)21-s + 1.39·23-s + 1.72i·25-s + (0.756 − 0.756i)27-s + (−1.11 + 1.11i)29-s − 0.976i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(2.45799 + 1.37308i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.45799 + 1.37308i\) |
\(L(3)\) |
|
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 | \( 1 + (-4.63 - 4.63i)T + 81iT^{2} \) |
| 5 | \( 1 + (-29.2 - 29.2i)T + 625iT^{2} \) |
| 7 | \( 1 - 59.6T + 2.40e3T^{2} \) |
| 11 | \( 1 + (-18.0 + 18.0i)T - 1.46e4iT^{2} \) |
| 13 | \( 1 + (50.7 - 50.7i)T - 2.85e4iT^{2} \) |
| 17 | \( 1 + 223.T + 8.35e4T^{2} \) |
| 19 | \( 1 + (14.7 + 14.7i)T + 1.30e5iT^{2} \) |
| 23 | \( 1 - 739.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (938. - 938. i)T - 7.07e5iT^{2} \) |
| 31 | \( 1 + 938. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + (263. + 263. i)T + 1.87e6iT^{2} \) |
| 41 | \( 1 - 248. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + (1.03e3 - 1.03e3i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 - 2.01e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + (833. + 833. i)T + 7.89e6iT^{2} \) |
| 59 | \( 1 + (-2.22e3 + 2.22e3i)T - 1.21e7iT^{2} \) |
| 61 | \( 1 + (-341. + 341. i)T - 1.38e7iT^{2} \) |
| 67 | \( 1 + (4.84e3 + 4.84e3i)T + 2.01e7iT^{2} \) |
| 71 | \( 1 + 4.18e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 9.07e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 735. iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (1.44e3 + 1.44e3i)T + 4.74e7iT^{2} \) |
| 89 | \( 1 + 5.07e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 2.52e3T + 8.85e7T^{2} \) |
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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
−13.06061102993930927017448552086, −11.41140344450675096897426069243, −10.75433360874628963663451716037, −9.605224806249718561114671053583, −8.841258280701176291665306593099, −7.30446778612276831453899455142, −6.20394680919539063231416475097, −4.77915472155862230295851276338, −3.16841777510735021781998520329, −1.85767291709422381269939923441,
1.32858157613355840910566388927, 2.25822344240352098709012069140, 4.69738619504676668393352379008, 5.48028573390952417312381799387, 7.17699115752864661725672037547, 8.390863681781978392379925688746, 9.007806950036654102483244853846, 10.31193360544279554885902473847, 11.53008911107249943071360574514, 12.83949048829993660969985780790