| L(s) = 1 | + (−1.33 + 0.458i)2-s + (0.594 − 1.62i)3-s + (1.57 − 1.22i)4-s + (−1.29 + 0.592i)5-s + (−0.0494 + 2.44i)6-s + (−3.42 − 3.59i)7-s + (−1.55 + 2.36i)8-s + (−2.29 − 1.93i)9-s + (1.46 − 1.38i)10-s + (−3.18 − 0.303i)11-s + (−1.05 − 3.29i)12-s + (6.15 + 1.18i)13-s + (6.23 + 3.23i)14-s + (0.192 + 2.46i)15-s + (0.992 − 3.87i)16-s + (0.341 − 0.661i)17-s + ⋯ |
| L(s) = 1 | + (−0.946 + 0.324i)2-s + (0.343 − 0.939i)3-s + (0.789 − 0.613i)4-s + (−0.579 + 0.264i)5-s + (−0.0201 + 0.999i)6-s + (−1.29 − 1.35i)7-s + (−0.548 + 0.836i)8-s + (−0.764 − 0.644i)9-s + (0.462 − 0.438i)10-s + (−0.959 − 0.0916i)11-s + (−0.304 − 0.952i)12-s + (1.70 + 0.328i)13-s + (1.66 + 0.865i)14-s + (0.0498 + 0.635i)15-s + (0.248 − 0.968i)16-s + (0.0827 − 0.160i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.623 - 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0145231 + 0.0301756i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0145231 + 0.0301756i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.33 - 0.458i)T \) |
| 3 | \( 1 + (-0.594 + 1.62i)T \) |
| 67 | \( 1 + (7.16 - 3.96i)T \) |
| good | 5 | \( 1 + (1.29 - 0.592i)T + (3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (3.42 + 3.59i)T + (-0.333 + 6.99i)T^{2} \) |
| 11 | \( 1 + (3.18 + 0.303i)T + (10.8 + 2.08i)T^{2} \) |
| 13 | \( 1 + (-6.15 - 1.18i)T + (12.0 + 4.83i)T^{2} \) |
| 17 | \( 1 + (-0.341 + 0.661i)T + (-9.86 - 13.8i)T^{2} \) |
| 19 | \( 1 + (0.192 - 0.202i)T + (-0.904 - 18.9i)T^{2} \) |
| 23 | \( 1 + (6.89 - 2.75i)T + (16.6 - 15.8i)T^{2} \) |
| 29 | \( 1 + (-5.39 - 3.11i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.554 - 2.87i)T + (-28.7 + 11.5i)T^{2} \) |
| 37 | \( 1 + (2.95 + 5.11i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.97 + 0.141i)T + (40.8 - 3.89i)T^{2} \) |
| 43 | \( 1 + (-4.43 - 6.89i)T + (-17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + (6.45 + 5.07i)T + (11.0 + 45.6i)T^{2} \) |
| 53 | \( 1 + (-0.671 + 1.04i)T + (-22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-1.33 - 1.54i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (5.19 - 0.496i)T + (59.8 - 11.5i)T^{2} \) |
| 71 | \( 1 + (-10.0 + 5.16i)T + (41.1 - 57.8i)T^{2} \) |
| 73 | \( 1 + (16.2 - 1.55i)T + (71.6 - 13.8i)T^{2} \) |
| 79 | \( 1 + (-2.03 + 0.703i)T + (62.0 - 48.8i)T^{2} \) |
| 83 | \( 1 + (-3.86 + 5.43i)T + (-27.1 - 78.4i)T^{2} \) |
| 89 | \( 1 + (15.2 - 2.19i)T + (85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-2.13 - 3.69i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.659925981079453563414544901948, −8.670886635744893169376031431309, −7.86667709295923113390025583950, −7.29021376878305557331157063149, −6.51660331318653951727294722226, −5.85939455077013997786765021711, −3.80959981155868076414687856499, −2.99100701431321646102916001615, −1.33923168275404141291576667990, −0.02213926693719635002237925879,
2.40373336416543418116922836999, 3.20951728446160047550371643553, 4.13866112511817612711957550661, 5.76618018766294335105096191123, 6.32169208908772079289049229443, 8.030361536851953134425155871564, 8.365913221220810979845908922592, 9.095090188864194469024666371558, 9.978928401469030707260855139617, 10.45921027162350670863503194154
