| L(s) = 1 | + (1.40 − 0.196i)2-s + (−0.739 − 1.56i)3-s + (1.92 − 0.551i)4-s + (−0.405 − 0.266i)5-s + (−1.34 − 2.04i)6-s + (0.678 − 0.639i)7-s + (2.58 − 1.15i)8-s + (−1.90 + 2.31i)9-s + (−0.621 − 0.293i)10-s + (1.65 − 3.30i)11-s + (−2.28 − 2.60i)12-s + (−1.97 − 0.852i)13-s + (0.823 − 1.02i)14-s + (−0.117 + 0.833i)15-s + (3.39 − 2.12i)16-s + (−0.154 + 0.184i)17-s + ⋯ |
| L(s) = 1 | + (0.990 − 0.139i)2-s + (−0.427 − 0.904i)3-s + (0.961 − 0.275i)4-s + (−0.181 − 0.119i)5-s + (−0.548 − 0.835i)6-s + (0.256 − 0.241i)7-s + (0.913 − 0.406i)8-s + (−0.635 + 0.772i)9-s + (−0.196 − 0.0929i)10-s + (0.499 − 0.995i)11-s + (−0.659 − 0.751i)12-s + (−0.548 − 0.236i)13-s + (0.220 − 0.275i)14-s + (−0.0304 + 0.215i)15-s + (0.847 − 0.530i)16-s + (−0.0374 + 0.0446i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.180 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.49974 - 1.80038i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.49974 - 1.80038i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.40 + 0.196i)T \) |
| 3 | \( 1 + (0.739 + 1.56i)T \) |
| good | 5 | \( 1 + (0.405 + 0.266i)T + (1.98 + 4.59i)T^{2} \) |
| 7 | \( 1 + (-0.678 + 0.639i)T + (0.407 - 6.98i)T^{2} \) |
| 11 | \( 1 + (-1.65 + 3.30i)T + (-6.56 - 8.82i)T^{2} \) |
| 13 | \( 1 + (1.97 + 0.852i)T + (8.92 + 9.45i)T^{2} \) |
| 17 | \( 1 + (0.154 - 0.184i)T + (-2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (-2.97 + 2.49i)T + (3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (0.582 - 0.617i)T + (-1.33 - 22.9i)T^{2} \) |
| 29 | \( 1 + (7.65 + 0.894i)T + (28.2 + 6.68i)T^{2} \) |
| 31 | \( 1 + (2.31 - 9.77i)T + (-27.7 - 13.9i)T^{2} \) |
| 37 | \( 1 + (-6.33 - 1.11i)T + (34.7 + 12.6i)T^{2} \) |
| 41 | \( 1 + (-6.78 + 5.05i)T + (11.7 - 39.2i)T^{2} \) |
| 43 | \( 1 + (-0.111 - 1.90i)T + (-42.7 + 4.99i)T^{2} \) |
| 47 | \( 1 + (6.24 - 1.48i)T + (42.0 - 21.0i)T^{2} \) |
| 53 | \( 1 + (-0.422 + 0.731i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.02 - 2.03i)T + (-35.2 + 47.3i)T^{2} \) |
| 61 | \( 1 + (-3.82 - 1.14i)T + (50.9 + 33.5i)T^{2} \) |
| 67 | \( 1 + (-13.3 + 1.56i)T + (65.1 - 15.4i)T^{2} \) |
| 71 | \( 1 + (-4.03 + 1.46i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (1.45 + 0.530i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-12.3 - 9.21i)T + (22.6 + 75.6i)T^{2} \) |
| 83 | \( 1 + (-3.38 - 2.52i)T + (23.8 + 79.5i)T^{2} \) |
| 89 | \( 1 + (3.16 - 8.69i)T + (-68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (13.0 - 8.58i)T + (38.4 - 89.0i)T^{2} \) |
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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
−10.92660009542824505241664309672, −9.564574406327489741153498328197, −8.226437084987120794955253116610, −7.45162378530700065726138694445, −6.61430182146315716930918562536, −5.72943997647921645933657806552, −4.93379953259972411973461560829, −3.66663895553842186557669852390, −2.45266719185129792722703571230, −1.02971007299898553317648183382,
2.10586165365187983389376004818, 3.57227863086426991917484729059, 4.31072663604345694445766601829, 5.24095593069232076677559765636, 6.00249250899182821366854123879, 7.11539540395141906683924974579, 7.921778628618135146302711923432, 9.424945162535065326337115831963, 9.886754271161662803537436747875, 11.27267370576464302808364067669
