| L(s) = 1 | + (−0.696 − 1.23i)2-s + (0.453 + 0.891i)3-s + (−1.03 + 1.71i)4-s + (0.328 − 2.21i)5-s + (0.780 − 1.17i)6-s + (−3.17 − 3.17i)7-s + (2.82 + 0.0740i)8-s + (−0.587 + 0.809i)9-s + (−2.95 + 1.13i)10-s + (1.22 + 1.68i)11-s + (−1.99 − 0.139i)12-s + (−0.339 − 2.14i)13-s + (−1.69 + 6.11i)14-s + (2.11 − 0.711i)15-s + (−1.87 − 3.53i)16-s + (−2.26 − 1.15i)17-s + ⋯ |
| L(s) = 1 | + (−0.492 − 0.870i)2-s + (0.262 + 0.514i)3-s + (−0.515 + 0.857i)4-s + (0.146 − 0.989i)5-s + (0.318 − 0.481i)6-s + (−1.20 − 1.20i)7-s + (0.999 + 0.0261i)8-s + (−0.195 + 0.269i)9-s + (−0.933 + 0.359i)10-s + (0.368 + 0.507i)11-s + (−0.575 − 0.0402i)12-s + (−0.0942 − 0.594i)13-s + (−0.453 + 1.63i)14-s + (0.547 − 0.183i)15-s + (−0.469 − 0.882i)16-s + (−0.548 − 0.279i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.762 + 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.762 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.263710 - 0.718173i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.263710 - 0.718173i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.696 + 1.23i)T \) |
| 3 | \( 1 + (-0.453 - 0.891i)T \) |
| 5 | \( 1 + (-0.328 + 2.21i)T \) |
| good | 7 | \( 1 + (3.17 + 3.17i)T + 7iT^{2} \) |
| 11 | \( 1 + (-1.22 - 1.68i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.339 + 2.14i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (2.26 + 1.15i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (1.87 + 5.78i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.703 + 4.44i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (-1.37 - 0.445i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (1.68 - 0.546i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-8.87 + 1.40i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (-5.90 - 4.28i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (4.19 - 4.19i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.82 - 0.928i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-10.2 + 5.24i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-3.92 - 2.84i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (11.1 - 8.09i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-3.16 + 6.20i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-8.47 - 2.75i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (0.968 + 0.153i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-4.20 + 12.9i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-4.17 - 2.12i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (-9.07 - 12.4i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (4.76 + 9.35i)T + (-57.0 + 78.4i)T^{2} \) |
| 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
−11.14886092376296995225441552818, −10.29933482358773679562215015062, −9.535278697137122419422939818904, −8.960821588376148091961848025228, −7.79903848067795107350121790564, −6.64576268968205662475151217494, −4.74074910577752181409281173963, −4.04282483477202657553963667795, −2.69385161459448188897774309119, −0.63673319783351493057823249467,
2.18542650321685067780153385376, 3.68109717252213902788862569114, 5.81081699718037972101129778704, 6.26292151447578535458728468511, 7.12559823761301145000228498449, 8.280377585598242592252660995196, 9.216587068188416388935365431271, 9.860702390896759099221915048362, 11.05727417829458879548485452495, 12.15238095211731495327775010537
