| L(s) = 1 | + (0.623 + 0.781i)2-s + (−0.933 − 0.140i)3-s + (−0.222 + 0.974i)4-s + (−0.826 − 0.563i)5-s + (−0.472 − 0.817i)6-s + (−0.0334 + 0.0579i)7-s + (−0.900 + 0.433i)8-s + (−2.01 − 0.621i)9-s + (−0.0747 − 0.997i)10-s + (1.09 + 4.77i)11-s + (0.344 − 0.879i)12-s + (−0.403 + 5.38i)13-s + (−0.0661 + 0.00997i)14-s + (0.692 + 0.642i)15-s + (−0.900 − 0.433i)16-s + (−1.58 + 1.08i)17-s + ⋯ |
| L(s) = 1 | + (0.440 + 0.552i)2-s + (−0.539 − 0.0812i)3-s + (−0.111 + 0.487i)4-s + (−0.369 − 0.251i)5-s + (−0.192 − 0.333i)6-s + (−0.0126 + 0.0219i)7-s + (−0.318 + 0.153i)8-s + (−0.671 − 0.207i)9-s + (−0.0236 − 0.315i)10-s + (0.328 + 1.44i)11-s + (0.0995 − 0.253i)12-s + (−0.112 + 1.49i)13-s + (−0.0176 + 0.00266i)14-s + (0.178 + 0.165i)15-s + (−0.225 − 0.108i)16-s + (−0.384 + 0.262i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.875 - 0.483i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.875 - 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.202165 + 0.783390i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.202165 + 0.783390i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.623 - 0.781i)T \) |
| 5 | \( 1 + (0.826 + 0.563i)T \) |
| 43 | \( 1 + (-6.13 - 2.31i)T \) |
| good | 3 | \( 1 + (0.933 + 0.140i)T + (2.86 + 0.884i)T^{2} \) |
| 7 | \( 1 + (0.0334 - 0.0579i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.09 - 4.77i)T + (-9.91 + 4.77i)T^{2} \) |
| 13 | \( 1 + (0.403 - 5.38i)T + (-12.8 - 1.93i)T^{2} \) |
| 17 | \( 1 + (1.58 - 1.08i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (7.78 - 2.40i)T + (15.6 - 10.7i)T^{2} \) |
| 23 | \( 1 + (-2.10 + 1.95i)T + (1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (4.55 - 0.687i)T + (27.7 - 8.54i)T^{2} \) |
| 31 | \( 1 + (-3.88 + 9.90i)T + (-22.7 - 21.0i)T^{2} \) |
| 37 | \( 1 + (-0.345 - 0.598i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.670 + 0.840i)T + (-9.12 + 39.9i)T^{2} \) |
| 47 | \( 1 + (2.51 - 11.0i)T + (-42.3 - 20.3i)T^{2} \) |
| 53 | \( 1 + (0.387 + 5.16i)T + (-52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (4.37 + 2.10i)T + (36.7 + 46.1i)T^{2} \) |
| 61 | \( 1 + (-2.11 - 5.38i)T + (-44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (-10.7 + 3.32i)T + (55.3 - 37.7i)T^{2} \) |
| 71 | \( 1 + (7.29 + 6.77i)T + (5.30 + 70.8i)T^{2} \) |
| 73 | \( 1 + (1.06 - 14.1i)T + (-72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (-7.43 + 12.8i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.84 - 0.881i)T + (79.3 + 24.4i)T^{2} \) |
| 89 | \( 1 + (8.54 + 1.28i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 + (2.00 + 8.78i)T + (-87.3 + 42.0i)T^{2} \) |
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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
−11.67557441048032580428041932252, −10.92551131256833517436710605208, −9.534500225580882706325576983303, −8.781313946982477520300497062894, −7.68931883345354211552745094363, −6.65154072298351482942622678644, −6.06997991741386794295005132955, −4.60950103616500912181033286436, −4.13931547461272657137466569350, −2.18228079530358649076525123590,
0.45720546679863712824983504321, 2.68626745008799409369301586584, 3.63759245364381247005887044806, 5.02264254099067280288197501730, 5.82535344888527268209612139892, 6.76630741742809015931698209393, 8.256794623759359422320766845843, 8.874615116161147741931453312714, 10.43761165759124957830225778134, 10.86371286781637157729082029651
