| L(s) = 1 | + (−1.39 + 0.241i)2-s + (2.01 + 0.289i)3-s + (1.88 − 0.672i)4-s + (−0.680 − 2.31i)5-s + (−2.88 + 0.0825i)6-s + (0.800 + 0.923i)7-s + (−2.46 + 1.39i)8-s + (1.10 + 0.324i)9-s + (1.50 + 3.06i)10-s + (2.66 + 1.71i)11-s + (3.99 − 0.810i)12-s + (1.15 − 1.33i)13-s + (−1.33 − 1.09i)14-s + (−0.700 − 4.86i)15-s + (3.09 − 2.53i)16-s + (−6.70 + 3.06i)17-s + ⋯ |
| L(s) = 1 | + (−0.985 + 0.170i)2-s + (1.16 + 0.167i)3-s + (0.941 − 0.336i)4-s + (−0.304 − 1.03i)5-s + (−1.17 + 0.0337i)6-s + (0.302 + 0.349i)7-s + (−0.870 + 0.491i)8-s + (0.368 + 0.108i)9-s + (0.476 + 0.969i)10-s + (0.804 + 0.517i)11-s + (1.15 − 0.233i)12-s + (0.321 − 0.370i)13-s + (−0.357 − 0.292i)14-s + (−0.180 − 1.25i)15-s + (0.773 − 0.633i)16-s + (−1.62 + 0.742i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0280i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0280i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.890773 - 0.0125106i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.890773 - 0.0125106i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 - 0.241i)T \) |
| 23 | \( 1 + (1.31 + 4.61i)T \) |
| good | 3 | \( 1 + (-2.01 - 0.289i)T + (2.87 + 0.845i)T^{2} \) |
| 5 | \( 1 + (0.680 + 2.31i)T + (-4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (-0.800 - 0.923i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-2.66 - 1.71i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-1.15 + 1.33i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (6.70 - 3.06i)T + (11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (2.65 - 5.81i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (1.47 + 3.23i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (3.77 - 0.543i)T + (29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (0.727 - 2.47i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-8.68 + 2.55i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.0321 + 0.223i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 7.98iT - 47T^{2} \) |
| 53 | \( 1 + (2.04 - 1.77i)T + (7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-10.9 - 9.49i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (2.87 - 0.414i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (1.08 - 0.697i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (3.55 + 5.53i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-1.05 + 2.30i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-9.02 + 10.4i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (1.24 + 0.366i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-12.2 - 1.76i)T + (85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (3.02 + 10.2i)T + (-81.6 + 52.4i)T^{2} \) |
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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
−14.49729180958536418102575646777, −12.92437928695131807319304623892, −11.91052725937605609058909964458, −10.54211709448443413200611643972, −9.153583512768861933902882090930, −8.670607093931795776096905212190, −7.87940314312134776951237228226, −6.17512207426576758850663441344, −4.14890474378642539887199520710, −2.03460185771082469845000676412,
2.34978872640513454812384083733, 3.66368933879426304439164816338, 6.66565555257721418556578399061, 7.41794780500982300393873393228, 8.713975248620342972059080076129, 9.329559463214510093903852161046, 11.10268192437055297660899996880, 11.26401549988804168481554062019, 13.21524260494755438515664211891, 14.22848317634812002978946237045
