L(s) = 1 | + 2-s + (−1.70 − 0.292i)3-s + 4-s + (−2 + i)5-s + (−1.70 − 0.292i)6-s + (0.121 + 0.292i)7-s + 8-s + (2.82 + i)9-s + (−2 + i)10-s + (−5.12 − 2.12i)11-s + (−1.70 − 0.292i)12-s + (−1.58 − 1.58i)13-s + (0.121 + 0.292i)14-s + (3.70 − 1.12i)15-s + 16-s + (−3 − 2.82i)17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.985 − 0.169i)3-s + 0.5·4-s + (−0.894 + 0.447i)5-s + (−0.696 − 0.119i)6-s + (0.0458 + 0.110i)7-s + 0.353·8-s + (0.942 + 0.333i)9-s + (−0.632 + 0.316i)10-s + (−1.54 − 0.639i)11-s + (−0.492 − 0.0845i)12-s + (−0.439 − 0.439i)13-s + (0.0324 + 0.0782i)14-s + (0.957 − 0.289i)15-s + 0.250·16-s + (−0.727 − 0.685i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0153i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0153i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (1.70 + 0.292i)T \) |
| 5 | \( 1 + (2 - i)T \) |
| 17 | \( 1 + (3 + 2.82i)T \) |
good | 7 | \( 1 + (-0.121 - 0.292i)T + (-4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (5.12 + 2.12i)T + (7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (1.58 + 1.58i)T + 13iT^{2} \) |
| 19 | \( 1 + (5.24 - 5.24i)T - 19iT^{2} \) |
| 23 | \( 1 + (2.70 - 1.12i)T + (16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (4.12 - 1.70i)T + (20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + (-0.707 - 1.70i)T + (-21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (-1.29 + 3.12i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (3.12 - 7.53i)T + (-28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 - 0.343iT - 43T^{2} \) |
| 47 | \( 1 + (0.171 + 0.171i)T + 47iT^{2} \) |
| 53 | \( 1 + 6.82iT - 53T^{2} \) |
| 59 | \( 1 + (3.58 - 3.58i)T - 59iT^{2} \) |
| 61 | \( 1 + (-10.9 - 4.53i)T + (43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + (1.58 + 1.58i)T + 67iT^{2} \) |
| 71 | \( 1 + (-10.3 + 4.29i)T + (50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (-3.36 + 8.12i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (10.1 + 4.19i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 - 0.343iT - 83T^{2} \) |
| 89 | \( 1 - 14.1iT - 89T^{2} \) |
| 97 | \( 1 + (8.53 + 3.53i)T + (68.5 + 68.5i)T^{2} \) |
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
−10.74591360192952172032333264253, −10.08466558854534085552539742003, −8.266445061117708127835835429440, −7.59984017878792850319767191033, −6.64608398607402949666195965721, −5.66922392703823819566453844641, −4.85255032053737314025879887164, −3.74235296179905103760127929624, −2.39498216652732731623513901790, 0,
2.27395752973344080958453604511, 4.10503141518493789751057548896, 4.62736714202855921332598083703, 5.52328304204547191191630977329, 6.73285117076420804052892889610, 7.46766135492449091093923773863, 8.526047761184092982543663488265, 9.870562083984329573685736395240, 10.83916920615807432149305746024