| L(s) = 1 | + (1.33 − 0.457i)2-s + (0.0516 − 1.73i)3-s + (1.58 − 1.22i)4-s + (−0.342 − 0.939i)5-s + (−0.722 − 2.34i)6-s + (−1.57 + 0.277i)7-s + (1.55 − 2.36i)8-s + (−2.99 − 0.178i)9-s + (−0.887 − 1.10i)10-s + (−5.31 − 1.93i)11-s + (−2.03 − 2.80i)12-s + (1.47 + 1.23i)13-s + (−1.97 + 1.08i)14-s + (−1.64 + 0.543i)15-s + (1.00 − 3.87i)16-s + (2.22 + 1.28i)17-s + ⋯ |
| L(s) = 1 | + (0.946 − 0.323i)2-s + (0.0298 − 0.999i)3-s + (0.790 − 0.611i)4-s + (−0.152 − 0.420i)5-s + (−0.294 − 0.955i)6-s + (−0.594 + 0.104i)7-s + (0.550 − 0.834i)8-s + (−0.998 − 0.0596i)9-s + (−0.280 − 0.348i)10-s + (−1.60 − 0.583i)11-s + (−0.587 − 0.808i)12-s + (0.408 + 0.343i)13-s + (−0.528 + 0.291i)14-s + (−0.424 + 0.140i)15-s + (0.251 − 0.967i)16-s + (0.540 + 0.311i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.840234 - 1.98301i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.840234 - 1.98301i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.33 + 0.457i)T \) |
| 3 | \( 1 + (-0.0516 + 1.73i)T \) |
| 5 | \( 1 + (0.342 + 0.939i)T \) |
| good | 7 | \( 1 + (1.57 - 0.277i)T + (6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (5.31 + 1.93i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.47 - 1.23i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.22 - 1.28i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-7.30 + 4.21i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.115 + 0.654i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-5.35 - 6.37i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-2.27 - 0.401i)T + (29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-3.72 + 6.45i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.51 + 1.80i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (1.50 - 4.12i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (1.90 + 10.8i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 2.08iT - 53T^{2} \) |
| 59 | \( 1 + (1.28 - 0.467i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-2.46 - 13.9i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-6.77 + 8.07i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.0169 + 0.0293i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (0.300 + 0.520i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.74 - 10.4i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (10.9 - 9.15i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (6.64 - 3.83i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.77 - 0.647i)T + (74.3 + 62.3i)T^{2} \) |
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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
−10.83098447128367869073555449142, −9.753854104068836762732730497496, −8.552447829043025110443890081536, −7.58766941840191249187905372873, −6.74591213566516941884912121586, −5.68119686317637861457063193966, −5.06975148962704034149279025672, −3.37272536209149944002537706827, −2.60102022877369595325725433993, −0.942354738365453733464455984272,
2.78826090388643669372408134330, 3.35909227506564189068914124086, 4.61113553318825433150788456339, 5.44765292314700544082082517878, 6.28349798277589359269038471082, 7.65348490767743783428722483214, 8.106187000084678060976639778635, 9.830449846295943958753386527844, 10.15673745548441353716792087189, 11.21552785709097477453532317068
