L(s) = 1 | + (−0.579 + 1.78i)2-s + (1.43 − 1.04i)3-s + (−1.22 − 0.893i)4-s + (0.309 + 0.951i)5-s + (1.03 + 3.17i)6-s + (−3.44 − 2.50i)7-s + (−0.729 + 0.529i)8-s + (0.0492 − 0.151i)9-s − 1.87·10-s + (2.44 − 2.24i)11-s − 2.70·12-s + (0.420 − 1.29i)13-s + (6.46 − 4.69i)14-s + (1.43 + 1.04i)15-s + (−1.46 − 4.49i)16-s + (0.648 + 1.99i)17-s + ⋯ |
L(s) = 1 | + (−0.409 + 1.26i)2-s + (0.830 − 0.603i)3-s + (−0.614 − 0.446i)4-s + (0.138 + 0.425i)5-s + (0.420 + 1.29i)6-s + (−1.30 − 0.945i)7-s + (−0.257 + 0.187i)8-s + (0.0164 − 0.0505i)9-s − 0.593·10-s + (0.737 − 0.675i)11-s − 0.779·12-s + (0.116 − 0.358i)13-s + (1.72 − 1.25i)14-s + (0.371 + 0.269i)15-s + (−0.365 − 1.12i)16-s + (0.157 + 0.484i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.451 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.705060 + 0.433197i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.705060 + 0.433197i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.309 - 0.951i)T \) |
| 11 | \( 1 + (-2.44 + 2.24i)T \) |
good | 2 | \( 1 + (0.579 - 1.78i)T + (-1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (-1.43 + 1.04i)T + (0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 + (3.44 + 2.50i)T + (2.16 + 6.65i)T^{2} \) |
| 13 | \( 1 + (-0.420 + 1.29i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.648 - 1.99i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (0.489 - 0.355i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + 4.39T + 23T^{2} \) |
| 29 | \( 1 + (-5.36 - 3.89i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (0.678 - 2.08i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (4.99 + 3.62i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (5.99 - 4.35i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 12.6T + 43T^{2} \) |
| 47 | \( 1 + (-2.48 + 1.80i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (2.05 - 6.32i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (9.73 + 7.07i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (1.75 + 5.41i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 - 9.86T + 67T^{2} \) |
| 71 | \( 1 + (1.61 + 4.98i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.584 - 0.424i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-1.75 + 5.39i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (0.294 + 0.906i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 - 1.24T + 89T^{2} \) |
| 97 | \( 1 + (-3.56 + 10.9i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−15.69139696328869100198401947513, −14.25273394229447742968855979751, −13.86489649015460187158009118873, −12.52524046078440788046210220228, −10.58063373073910852841072329383, −9.191849421381954511482485537377, −8.059144934006416348291466706319, −7.00592571643802829310508230906, −6.14051789981519436023125711652, −3.28912258884919088878562794116,
2.54691041817812437494498123915, 3.91067239105012126372886748289, 6.32566307842299318672387587569, 8.761500535745882288904193326762, 9.433855269585725856846801809122, 10.06498780138290884972709111709, 11.90112766681534973153899704775, 12.44772389098754262565045882646, 13.86222345423448261895975376290, 15.28474377918376331182109105710