| L(s) = 1 | + (−1.39 − 0.236i)2-s + (1.65 + 0.501i)3-s + (1.88 + 0.660i)4-s + (1.24 − 0.567i)5-s + (−2.19 − 1.09i)6-s + (2.47 + 2.59i)7-s + (−2.47 − 1.36i)8-s + (2.49 + 1.66i)9-s + (−1.86 + 0.496i)10-s + (4.55 + 0.435i)11-s + (2.79 + 2.04i)12-s + (−3.40 − 0.656i)13-s + (−2.83 − 4.20i)14-s + (2.34 − 0.318i)15-s + (3.12 + 2.49i)16-s + (0.522 − 1.01i)17-s + ⋯ |
| L(s) = 1 | + (−0.985 − 0.167i)2-s + (0.957 + 0.289i)3-s + (0.943 + 0.330i)4-s + (0.555 − 0.253i)5-s + (−0.895 − 0.445i)6-s + (0.935 + 0.981i)7-s + (−0.875 − 0.483i)8-s + (0.832 + 0.553i)9-s + (−0.590 + 0.157i)10-s + (1.37 + 0.131i)11-s + (0.807 + 0.589i)12-s + (−0.944 − 0.182i)13-s + (−0.758 − 1.12i)14-s + (0.605 − 0.0821i)15-s + (0.781 + 0.623i)16-s + (0.126 − 0.245i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.72590 + 0.415580i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.72590 + 0.415580i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 + 0.236i)T \) |
| 3 | \( 1 + (-1.65 - 0.501i)T \) |
| 67 | \( 1 + (2.62 - 7.75i)T \) |
| good | 5 | \( 1 + (-1.24 + 0.567i)T + (3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (-2.47 - 2.59i)T + (-0.333 + 6.99i)T^{2} \) |
| 11 | \( 1 + (-4.55 - 0.435i)T + (10.8 + 2.08i)T^{2} \) |
| 13 | \( 1 + (3.40 + 0.656i)T + (12.0 + 4.83i)T^{2} \) |
| 17 | \( 1 + (-0.522 + 1.01i)T + (-9.86 - 13.8i)T^{2} \) |
| 19 | \( 1 + (-1.53 + 1.61i)T + (-0.904 - 18.9i)T^{2} \) |
| 23 | \( 1 + (7.12 - 2.85i)T + (16.6 - 15.8i)T^{2} \) |
| 29 | \( 1 + (-5.69 - 3.29i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.80 + 9.37i)T + (-28.7 + 11.5i)T^{2} \) |
| 37 | \( 1 + (2.75 + 4.77i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.80 - 0.133i)T + (40.8 - 3.89i)T^{2} \) |
| 43 | \( 1 + (1.26 + 1.96i)T + (-17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + (5.51 + 4.34i)T + (11.0 + 45.6i)T^{2} \) |
| 53 | \( 1 + (5.38 - 8.38i)T + (-22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (2.65 + 3.06i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (6.02 - 0.575i)T + (59.8 - 11.5i)T^{2} \) |
| 71 | \( 1 + (-13.1 + 6.77i)T + (41.1 - 57.8i)T^{2} \) |
| 73 | \( 1 + (-10.5 + 1.00i)T + (71.6 - 13.8i)T^{2} \) |
| 79 | \( 1 + (-10.0 + 3.46i)T + (62.0 - 48.8i)T^{2} \) |
| 83 | \( 1 + (-0.687 + 0.965i)T + (-27.1 - 78.4i)T^{2} \) |
| 89 | \( 1 + (1.90 - 0.274i)T + (85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (1.27 + 2.21i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.787249921860916201168091473150, −9.517668497266891105790724775057, −8.790640852694526772266270641046, −8.003544848802496910456417894629, −7.27011560491296539991472285394, −6.05004562536369758551894783291, −4.93954500157075805324133860130, −3.61815326633879825047672329285, −2.29248497778197559741471011087, −1.66171739161788637486629992461,
1.29039358852302488974338233474, 2.09778719056444134467715383197, 3.50393450514081860763349914875, 4.71017731256751027640112111065, 6.39421465799033670478120245720, 6.82677878658256519574154350439, 7.944747337254604682380746160008, 8.275589591142855751948304218933, 9.433876072490343336355870537330, 9.970277543334884829069131253310
