| L(s) = 1 | + (−0.230 + 0.399i)2-s + (−1.5 − 2.59i)3-s + (3.89 + 6.74i)4-s + (−2.5 + 4.33i)5-s + 1.38·6-s + (−18.0 − 4.01i)7-s − 7.28·8-s + (−4.5 + 7.79i)9-s + (−1.15 − 1.99i)10-s + (12.1 + 21.0i)11-s + (11.6 − 20.2i)12-s − 66.1·13-s + (5.77 − 6.29i)14-s + 15.0·15-s + (−29.4 + 51.0i)16-s + (9.80 + 16.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.0815 + 0.141i)2-s + (−0.288 − 0.499i)3-s + (0.486 + 0.842i)4-s + (−0.223 + 0.387i)5-s + 0.0941·6-s + (−0.976 − 0.216i)7-s − 0.321·8-s + (−0.166 + 0.288i)9-s + (−0.0364 − 0.0631i)10-s + (0.333 + 0.577i)11-s + (0.280 − 0.486i)12-s − 1.41·13-s + (0.110 − 0.120i)14-s + 0.258·15-s + (−0.460 + 0.797i)16-s + (0.139 + 0.242i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.778 - 0.627i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.230786 + 0.653762i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.230786 + 0.653762i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.5 + 2.59i)T \) |
| 5 | \( 1 + (2.5 - 4.33i)T \) |
| 7 | \( 1 + (18.0 + 4.01i)T \) |
| good | 2 | \( 1 + (0.230 - 0.399i)T + (-4 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-12.1 - 21.0i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 66.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-9.80 - 16.9i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (32.3 - 56.0i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (73.7 - 127. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 166.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (94.9 + 164. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-37.9 + 65.7i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 186.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 110.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-190. + 329. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-356. - 616. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (154. + 267. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (244. - 424. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (378. + 656. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 58.0T + 3.57e5T^{2} \) |
| 73 | \( 1 + (29.0 + 50.2i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (432. - 749. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 829.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (546. - 946. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.11e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.48948450896528185189148139694, −12.32136461163913854415542450866, −12.03614989587658367542577609671, −10.56651387465889786342189784637, −9.416293096457112259777856248450, −7.76188509290860419658628922922, −7.13685255691154930016437160640, −6.05463426673717996628813611610, −3.94810715474166964051311053312, −2.44827642745509074848708831829,
0.38062364287870065718190354804, 2.74944926665125750190403787528, 4.65236850079770005725902168858, 5.92051975242460798125970903544, 6.96976543301309119281019245649, 8.841395851720505136300892440952, 9.784496409581685654229378491314, 10.60595914578778251779012577684, 11.81212197974781955975935547587, 12.58366651794877630592768198997
