| L(s) = 1 | + (0.171 − 1.96i)2-s + (0.120 + 0.0213i)4-s + (2.46 + 4.34i)5-s + (4.10 + 2.87i)7-s + (2.10 − 7.84i)8-s + (8.95 − 4.09i)10-s + (−6.12 − 2.22i)11-s + (1.03 + 11.8i)13-s + (6.35 − 7.56i)14-s + (−14.5 − 5.29i)16-s + (6.97 + 26.0i)17-s + (11.5 − 6.66i)19-s + (0.205 + 0.577i)20-s + (−5.42 + 11.6i)22-s + (27.6 − 19.3i)23-s + ⋯ |
| L(s) = 1 | + (0.0858 − 0.980i)2-s + (0.0302 + 0.00532i)4-s + (0.493 + 0.869i)5-s + (0.587 + 0.411i)7-s + (0.262 − 0.980i)8-s + (0.895 − 0.409i)10-s + (−0.556 − 0.202i)11-s + (0.0796 + 0.909i)13-s + (0.453 − 0.540i)14-s + (−0.909 − 0.331i)16-s + (0.410 + 1.53i)17-s + (0.607 − 0.350i)19-s + (0.0102 + 0.0288i)20-s + (−0.246 + 0.528i)22-s + (1.20 − 0.841i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 + 0.424i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.905 + 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.28605 - 0.509383i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.28605 - 0.509383i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-2.46 - 4.34i)T \) |
| good | 2 | \( 1 + (-0.171 + 1.96i)T + (-3.93 - 0.694i)T^{2} \) |
| 7 | \( 1 + (-4.10 - 2.87i)T + (16.7 + 46.0i)T^{2} \) |
| 11 | \( 1 + (6.12 + 2.22i)T + (92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (-1.03 - 11.8i)T + (-166. + 29.3i)T^{2} \) |
| 17 | \( 1 + (-6.97 - 26.0i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (-11.5 + 6.66i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-27.6 + 19.3i)T + (180. - 497. i)T^{2} \) |
| 29 | \( 1 + (-26.8 - 31.9i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (-0.723 + 4.10i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (9.85 + 36.7i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (-49.9 - 41.8i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (9.03 + 19.3i)T + (-1.18e3 + 1.41e3i)T^{2} \) |
| 47 | \( 1 + (71.4 + 50.0i)T + (755. + 2.07e3i)T^{2} \) |
| 53 | \( 1 + (36.7 + 36.7i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-9.93 - 27.2i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-0.481 - 2.73i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (22.4 - 1.96i)T + (4.42e3 - 779. i)T^{2} \) |
| 71 | \( 1 + (-24.7 + 42.8i)T + (-2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (7.26 - 27.1i)T + (-4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-56.3 - 67.1i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-11.8 - 1.03i)T + (6.78e3 + 1.19e3i)T^{2} \) |
| 89 | \( 1 + (-56.3 + 32.5i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (1.05 - 0.492i)T + (6.04e3 - 7.20e3i)T^{2} \) |
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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
−10.93687959647451515789038372977, −10.43423510408653088239656045014, −9.442633161549587855826922831269, −8.341464946502357856658017642907, −7.09747678833256604412444247510, −6.30763080331141308981189995691, −5.00434931057994865965998140871, −3.57780586663949291882773167212, −2.57228075125704336611113170437, −1.54059712375787020739709244714,
1.14363957898445737635440908730, 2.80290006929578741227546878941, 4.84628385424969944403746900802, 5.21950192280009651939680668737, 6.30106255338624325708594647981, 7.63434274535718679928226337286, 7.892506378134995099540029936264, 9.147098285812991671694799501875, 10.07421166317641514025636521615, 11.13379965164587602210506856595
