| L(s) = 1 | + (1.46 + 2.54i)2-s + (1.5 + 0.866i)3-s + (−2.30 + 3.99i)4-s + (1.93 − 1.11i)5-s + 5.08i·6-s + (−5.41 + 4.43i)7-s − 1.79·8-s + (1.5 + 2.59i)9-s + (5.68 + 3.28i)10-s + (0.802 − 1.38i)11-s + (−6.91 + 3.99i)12-s − 14.1i·13-s + (−19.2 − 7.25i)14-s + 3.87·15-s + (6.58 + 11.4i)16-s + (−14.5 − 8.39i)17-s + ⋯ |
| L(s) = 1 | + (0.733 + 1.27i)2-s + (0.5 + 0.288i)3-s + (−0.576 + 0.998i)4-s + (0.387 − 0.223i)5-s + 0.847i·6-s + (−0.773 + 0.633i)7-s − 0.224·8-s + (0.166 + 0.288i)9-s + (0.568 + 0.328i)10-s + (0.0729 − 0.126i)11-s + (−0.576 + 0.332i)12-s − 1.08i·13-s + (−1.37 − 0.518i)14-s + 0.258·15-s + (0.411 + 0.713i)16-s + (−0.855 − 0.494i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.285 - 0.958i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.285 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.31310 + 1.76059i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31310 + 1.76059i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.5 - 0.866i)T \) |
| 5 | \( 1 + (-1.93 + 1.11i)T \) |
| 7 | \( 1 + (5.41 - 4.43i)T \) |
| good | 2 | \( 1 + (-1.46 - 2.54i)T + (-2 + 3.46i)T^{2} \) |
| 11 | \( 1 + (-0.802 + 1.38i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 14.1iT - 169T^{2} \) |
| 17 | \( 1 + (14.5 + 8.39i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-20.5 + 11.8i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (9.88 + 17.1i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 22.0T + 841T^{2} \) |
| 31 | \( 1 + (-49.7 - 28.7i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-7.24 - 12.5i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 25.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 55.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-25.8 + 14.9i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (28.3 - 49.1i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (41.1 + 23.7i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (76.2 - 43.9i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (1.62 - 2.81i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 93.9T + 5.04e3T^{2} \) |
| 73 | \( 1 + (38.7 + 22.3i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-66.8 - 115. i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 21.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-85.1 + 49.1i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 155. iT - 9.40e3T^{2} \) |
| 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
−13.80085914761885238186030117396, −13.32510669924356878429838974230, −12.20164197896516188340411373522, −10.43509979297808123502110247930, −9.240071640971235850952490628964, −8.209788314868116996119816515330, −6.90335268906360280109273191234, −5.80507850613975737662186339584, −4.74291684592085954951294721199, −3.01478439294848408651575531741,
1.75538281298573417704955483174, 3.23728748454331934634103346772, 4.34773872808225377108728265567, 6.25302996379785328309498191879, 7.56934572824630948864423601980, 9.395133126838710989062187212675, 10.08928540006500624114136567983, 11.31788252206886480413889770678, 12.21536759895739734755186721468, 13.50182189860799765110985648397
