| L(s) = 1 | + (1.26 − 2.18i)3-s + (1.55 − 1.55i)5-s + (0.895 − 3.34i)7-s + (1.31 + 2.28i)9-s + (−1.53 + 0.410i)11-s + (−1.43 + 12.9i)13-s + (−1.43 − 5.36i)15-s + (−22.9 + 13.2i)17-s + (3.34 + 0.896i)19-s + (−6.16 − 6.16i)21-s + (−15.6 − 9.06i)23-s + 20.1i·25-s + 29.3·27-s + (4.04 − 7.01i)29-s + (30.2 − 30.2i)31-s + ⋯ |
| L(s) = 1 | + (0.420 − 0.728i)3-s + (0.311 − 0.311i)5-s + (0.127 − 0.477i)7-s + (0.146 + 0.253i)9-s + (−0.139 + 0.0373i)11-s + (−0.110 + 0.993i)13-s + (−0.0958 − 0.357i)15-s + (−1.35 + 0.779i)17-s + (0.176 + 0.0471i)19-s + (−0.293 − 0.293i)21-s + (−0.682 − 0.394i)23-s + 0.805i·25-s + 1.08·27-s + (0.139 − 0.241i)29-s + (0.976 − 0.976i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 + 0.623i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.782 + 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.23906 - 0.433201i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23906 - 0.433201i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + (1.43 - 12.9i)T \) |
| good | 3 | \( 1 + (-1.26 + 2.18i)T + (-4.5 - 7.79i)T^{2} \) |
| 5 | \( 1 + (-1.55 + 1.55i)T - 25iT^{2} \) |
| 7 | \( 1 + (-0.895 + 3.34i)T + (-42.4 - 24.5i)T^{2} \) |
| 11 | \( 1 + (1.53 - 0.410i)T + (104. - 60.5i)T^{2} \) |
| 17 | \( 1 + (22.9 - 13.2i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-3.34 - 0.896i)T + (312. + 180.5i)T^{2} \) |
| 23 | \( 1 + (15.6 + 9.06i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-4.04 + 7.01i)T + (-420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-30.2 + 30.2i)T - 961iT^{2} \) |
| 37 | \( 1 + (-38.5 + 10.3i)T + (1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 + (12.3 + 46.0i)T + (-1.45e3 + 840.5i)T^{2} \) |
| 43 | \( 1 + (41.4 - 23.9i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (60.4 + 60.4i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + 1.63T + 2.80e3T^{2} \) |
| 59 | \( 1 + (25.0 - 93.6i)T + (-3.01e3 - 1.74e3i)T^{2} \) |
| 61 | \( 1 + (20.8 + 36.1i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-6.99 - 26.1i)T + (-3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 + (93.5 + 25.0i)T + (4.36e3 + 2.52e3i)T^{2} \) |
| 73 | \( 1 + (73.5 + 73.5i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 82.5T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-34.5 + 34.5i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (-128. + 34.5i)T + (6.85e3 - 3.96e3i)T^{2} \) |
| 97 | \( 1 + (-67.6 - 18.1i)T + (8.14e3 + 4.70e3i)T^{2} \) |
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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
−14.93102712417139436016012079294, −13.63453136512134985368246221953, −13.18902305564899593270641223303, −11.76148304717834755291941857697, −10.37704001051259367668733995553, −8.930506773489446080304424370858, −7.71053674924629215627076885644, −6.45061471250303106890217184204, −4.45074275695045633550873981923, −1.96638210665143909173546114269,
2.91591220957254025338741639892, 4.76364717782584727816944738925, 6.47303444455778432593740833747, 8.257583884531959869662491416868, 9.486840481491417904183969539671, 10.43325008678547911706322715342, 11.82556473716011823944939123795, 13.20406122345746609178241772678, 14.40042327534088244940291434009, 15.39021741911587740972481381045
