| L(s) = 1 | + (2.25 + 0.107i)2-s + (0.597 − 1.62i)3-s + (3.07 + 0.293i)4-s + (−1.85 + 1.77i)5-s + (1.52 − 3.59i)6-s + (0.300 − 1.55i)7-s + (2.43 + 0.350i)8-s + (−2.28 − 1.94i)9-s + (−4.38 + 3.79i)10-s + (5.28 + 2.72i)11-s + (2.31 − 4.82i)12-s + (−5.29 + 1.01i)13-s + (0.844 − 3.48i)14-s + (1.77 + 4.08i)15-s + (−0.618 − 0.119i)16-s + (−1.57 + 3.45i)17-s + ⋯ |
| L(s) = 1 | + (1.59 + 0.0759i)2-s + (0.345 − 0.938i)3-s + (1.53 + 0.146i)4-s + (−0.831 + 0.792i)5-s + (0.621 − 1.46i)6-s + (0.113 − 0.589i)7-s + (0.861 + 0.123i)8-s + (−0.761 − 0.647i)9-s + (−1.38 + 1.20i)10-s + (1.59 + 0.821i)11-s + (0.668 − 1.39i)12-s + (−1.46 + 0.282i)13-s + (0.225 − 0.930i)14-s + (0.457 + 1.05i)15-s + (−0.154 − 0.0297i)16-s + (−0.382 + 0.837i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.397i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 + 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.51404 - 0.520615i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.51404 - 0.520615i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.597 + 1.62i)T \) |
| 23 | \( 1 + (-3.48 + 3.29i)T \) |
| good | 2 | \( 1 + (-2.25 - 0.107i)T + (1.99 + 0.190i)T^{2} \) |
| 5 | \( 1 + (1.85 - 1.77i)T + (0.237 - 4.99i)T^{2} \) |
| 7 | \( 1 + (-0.300 + 1.55i)T + (-6.49 - 2.60i)T^{2} \) |
| 11 | \( 1 + (-5.28 - 2.72i)T + (6.38 + 8.96i)T^{2} \) |
| 13 | \( 1 + (5.29 - 1.01i)T + (12.0 - 4.83i)T^{2} \) |
| 17 | \( 1 + (1.57 - 3.45i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (0.847 - 0.387i)T + (12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (0.741 + 7.76i)T + (-28.4 + 5.48i)T^{2} \) |
| 31 | \( 1 + (-0.873 - 0.687i)T + (7.30 + 30.1i)T^{2} \) |
| 37 | \( 1 + (0.0220 + 0.0752i)T + (-31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-5.40 - 5.67i)T + (-1.95 + 40.9i)T^{2} \) |
| 43 | \( 1 + (2.17 + 2.76i)T + (-10.1 + 41.7i)T^{2} \) |
| 47 | \( 1 + (-3.45 + 1.99i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.295 + 0.341i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (1.51 + 7.84i)T + (-54.7 + 21.9i)T^{2} \) |
| 61 | \( 1 + (-0.808 - 2.01i)T + (-44.1 + 42.0i)T^{2} \) |
| 67 | \( 1 + (4.19 + 8.13i)T + (-38.8 + 54.5i)T^{2} \) |
| 71 | \( 1 + (-2.81 + 4.37i)T + (-29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (3.26 + 7.14i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-3.40 - 1.17i)T + (62.0 + 48.8i)T^{2} \) |
| 83 | \( 1 + (-10.3 - 9.88i)T + (3.94 + 82.9i)T^{2} \) |
| 89 | \( 1 + (0.0508 + 0.353i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-6.82 + 1.65i)T + (86.2 - 44.4i)T^{2} \) |
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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
−12.27790317909898785332445498565, −11.94344769854556783919100259648, −10.95623185590930481096855929447, −9.341828927143767941178118195352, −7.71642465895453260259573425866, −6.92545350605000004738933764068, −6.37446261061444102823268034832, −4.52205904990971098494774227831, −3.69087726825502764326367911480, −2.30088475091348932245853180241,
2.82968367900942269677149720560, 3.95314419475775572072549542417, 4.79069806800660840327544801204, 5.60942874884496120921034633243, 7.15978703359073044175548883311, 8.720223532659715559192807450965, 9.311322826311384134751845709455, 11.01886114047465519401912975384, 11.82270035110662848888738123478, 12.30909903115787026204035929846
