| L(s) = 1 | + (1.70 − 1.70i)2-s + (2.12 − 2.12i)3-s − 1.82i·4-s + (−4.82 + 2i)5-s − 7.24i·6-s + (−3.82 + 9.24i)7-s + (3.70 + 3.70i)8-s − 8.99i·9-s + (−4.82 + 11.6i)10-s + (3.29 − 7.94i)11-s + (−3.87 − 3.87i)12-s − 14.8i·13-s + (9.24 + 22.3i)14-s + (−5.99 + 14.4i)15-s + 19.9·16-s + (−4.60 + 16.3i)17-s + ⋯ |
| L(s) = 1 | + (0.853 − 0.853i)2-s + (0.707 − 0.707i)3-s − 0.457i·4-s + (−0.965 + 0.400i)5-s − 1.20i·6-s + (−0.546 + 1.32i)7-s + (0.463 + 0.463i)8-s − 0.999i·9-s + (−0.482 + 1.16i)10-s + (0.299 − 0.722i)11-s + (−0.323 − 0.323i)12-s − 1.14i·13-s + (0.660 + 1.59i)14-s + (−0.399 + 0.965i)15-s + 1.24·16-s + (−0.270 + 0.962i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.49005 - 0.920066i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.49005 - 0.920066i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.12 + 2.12i)T \) |
| 17 | \( 1 + (4.60 - 16.3i)T \) |
| good | 2 | \( 1 + (-1.70 + 1.70i)T - 4iT^{2} \) |
| 5 | \( 1 + (4.82 - 2i)T + (17.6 - 17.6i)T^{2} \) |
| 7 | \( 1 + (3.82 - 9.24i)T + (-34.6 - 34.6i)T^{2} \) |
| 11 | \( 1 + (-3.29 + 7.94i)T + (-85.5 - 85.5i)T^{2} \) |
| 13 | \( 1 + 14.8iT - 169T^{2} \) |
| 19 | \( 1 + (12.8 + 12.8i)T + 361iT^{2} \) |
| 23 | \( 1 + (5.72 - 13.8i)T + (-374. - 374. i)T^{2} \) |
| 29 | \( 1 + (-18.3 + 7.61i)T + (594. - 594. i)T^{2} \) |
| 31 | \( 1 + (9.72 - 4.02i)T + (679. - 679. i)T^{2} \) |
| 37 | \( 1 + (-8.82 + 3.65i)T + (968. - 968. i)T^{2} \) |
| 41 | \( 1 + (-3.77 - 1.56i)T + (1.18e3 + 1.18e3i)T^{2} \) |
| 43 | \( 1 + (-22.5 + 22.5i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 - 87.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + (17.3 - 17.3i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (70.7 + 70.7i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (44.1 - 106. i)T + (-2.63e3 - 2.63e3i)T^{2} \) |
| 67 | \( 1 - 48.1T + 4.48e3T^{2} \) |
| 71 | \( 1 + (16.4 + 39.8i)T + (-3.56e3 + 3.56e3i)T^{2} \) |
| 73 | \( 1 + (-4.16 - 10.0i)T + (-3.76e3 + 3.76e3i)T^{2} \) |
| 79 | \( 1 + (-54.8 - 22.7i)T + (4.41e3 + 4.41e3i)T^{2} \) |
| 83 | \( 1 + (-86.7 + 86.7i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 0.159T + 7.92e3T^{2} \) |
| 97 | \( 1 + (20.4 + 49.2i)T + (-6.65e3 + 6.65e3i)T^{2} \) |
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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
−14.96760402256604545371530598178, −13.62983389787773520164796083874, −12.63358636706005115232610392882, −12.01848279135448282708526596821, −10.85224987712844918871598306841, −8.847220273071288707400714683596, −7.80161117378704168819170930107, −5.98226137279365479393565455070, −3.69310761888022620806299318843, −2.64063916672503786242798039205,
4.01512286321599360167434665994, 4.50265992540263428593831574780, 6.78745823455491607112742615716, 7.79374148899496923939897373627, 9.447085662121351815448036624255, 10.66005903984035983080605135259, 12.39237772613980672224576118221, 13.71822755337192363318460745779, 14.33217324936028210437595003920, 15.45244295735913469110902636265
