| L(s) = 1 | + (0.309 + 0.951i)2-s + (0.978 − 0.207i)3-s + (−0.809 + 0.587i)4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s + (0.577 + 0.257i)7-s + (−0.809 − 0.587i)8-s + (0.913 − 0.406i)9-s + (−0.978 − 0.207i)10-s + (−0.246 − 2.34i)11-s + (−0.669 + 0.743i)12-s + (4.74 + 5.27i)13-s + (−0.0661 + 0.629i)14-s + (−0.309 + 0.951i)15-s + (0.309 − 0.951i)16-s + (0.153 − 1.45i)17-s + ⋯ |
| L(s) = 1 | + (0.218 + 0.672i)2-s + (0.564 − 0.120i)3-s + (−0.404 + 0.293i)4-s + (−0.223 + 0.387i)5-s + (0.204 + 0.353i)6-s + (0.218 + 0.0972i)7-s + (−0.286 − 0.207i)8-s + (0.304 − 0.135i)9-s + (−0.309 − 0.0657i)10-s + (−0.0742 − 0.706i)11-s + (−0.193 + 0.214i)12-s + (1.31 + 1.46i)13-s + (−0.0176 + 0.168i)14-s + (−0.0797 + 0.245i)15-s + (0.0772 − 0.237i)16-s + (0.0371 − 0.353i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0757 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0757 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.38401 + 1.49311i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38401 + 1.49311i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.309 - 0.951i)T \) |
| 3 | \( 1 + (-0.978 + 0.207i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (-0.409 - 5.55i)T \) |
| good | 7 | \( 1 + (-0.577 - 0.257i)T + (4.68 + 5.20i)T^{2} \) |
| 11 | \( 1 + (0.246 + 2.34i)T + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (-4.74 - 5.27i)T + (-1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.153 + 1.45i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (3.50 - 3.88i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (-6.48 - 4.71i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.281 - 0.865i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (1.69 + 2.93i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.97 - 0.419i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (7.93 - 8.81i)T + (-4.49 - 42.7i)T^{2} \) |
| 47 | \( 1 + (-0.894 + 2.75i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-4.10 + 1.82i)T + (35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (-13.2 + 2.80i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + 8.58T + 61T^{2} \) |
| 67 | \( 1 + (0.848 - 1.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.14 - 1.39i)T + (47.5 - 52.7i)T^{2} \) |
| 73 | \( 1 + (0.547 + 5.20i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (-0.820 + 7.80i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (-1.34 - 0.286i)T + (75.8 + 33.7i)T^{2} \) |
| 89 | \( 1 + (-1.63 + 1.18i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-7.93 + 5.76i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.21313781056461854455194467253, −8.998401165748517524602882915624, −8.674886646031572282038341984052, −7.75399835468067596618000923001, −6.81753605020520014234810235708, −6.22602631688658282086389161625, −5.05509800218693552491778628084, −3.91074896074591440085957651876, −3.20946877484407577186007243982, −1.59698306353371749602699908495,
0.933350899114367528330630769963, 2.35932651069274629353055264717, 3.40482595420592585424193579593, 4.35566675524628514676334397248, 5.17081278546406106650057328040, 6.33400029574267585379166301618, 7.51125022217050875193238205932, 8.501108765416556924115395814001, 8.846493684935624277700972663614, 10.03104253467954532058572251089
