L(s) = 1 | + (−2.68 + 0.473i)2-s + (−2.95 + 0.527i)3-s + (3.22 − 1.17i)4-s + (4.68 + 5.58i)5-s + (7.67 − 2.81i)6-s + (−2.46 − 4.26i)7-s + (1.34 − 0.778i)8-s + (8.44 − 3.11i)9-s + (−15.2 − 12.7i)10-s − 7.13i·11-s + (−8.89 + 5.16i)12-s + (7.90 + 6.63i)13-s + (8.62 + 10.2i)14-s + (−16.7 − 14.0i)15-s + (−13.7 + 11.5i)16-s + (18.5 + 22.0i)17-s + ⋯ |
L(s) = 1 | + (−1.34 + 0.236i)2-s + (−0.984 + 0.175i)3-s + (0.805 − 0.293i)4-s + (0.936 + 1.11i)5-s + (1.27 − 0.469i)6-s + (−0.351 − 0.608i)7-s + (0.168 − 0.0973i)8-s + (0.938 − 0.346i)9-s + (−1.52 − 1.27i)10-s − 0.648i·11-s + (−0.741 + 0.430i)12-s + (0.608 + 0.510i)13-s + (0.615 + 0.733i)14-s + (−1.11 − 0.933i)15-s + (−0.859 + 0.721i)16-s + (1.08 + 1.29i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.332 - 0.943i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.332 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.304672 + 0.430548i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.304672 + 0.430548i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.95 - 0.527i)T \) |
| 19 | \( 1 + (18.9 - 0.0106i)T \) |
good | 2 | \( 1 + (2.68 - 0.473i)T + (3.75 - 1.36i)T^{2} \) |
| 5 | \( 1 + (-4.68 - 5.58i)T + (-4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (2.46 + 4.26i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + 7.13iT - 121T^{2} \) |
| 13 | \( 1 + (-7.90 - 6.63i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-18.5 - 22.0i)T + (-50.1 + 284. i)T^{2} \) |
| 23 | \( 1 + (-7.46 - 20.5i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (-2.84 - 7.81i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 + 31.7T + 961T^{2} \) |
| 37 | \( 1 + 14.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + (41.1 - 7.26i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (-61.1 - 22.2i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-23.6 - 64.9i)T + (-1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 + (43.7 + 7.71i)T + (2.63e3 + 960. i)T^{2} \) |
| 59 | \( 1 + (21.0 - 57.9i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-43.1 - 36.1i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (6.23 - 35.3i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-39.6 + 6.99i)T + (4.73e3 - 1.72e3i)T^{2} \) |
| 73 | \( 1 + (56.6 + 20.6i)T + (4.08e3 + 3.42e3i)T^{2} \) |
| 79 | \( 1 + (-82.4 + 69.1i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (61.2 - 35.3i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-19.1 - 52.5i)T + (-6.06e3 + 5.09e3i)T^{2} \) |
| 97 | \( 1 + (9.63 + 54.6i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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
−12.80661395169578611991972244009, −11.15659144042338090531230991849, −10.59581304758273944461405335239, −10.03842118549825114481915263750, −8.999014446820748672129719913113, −7.53975431301833676624976724021, −6.55911641593234931996769941206, −5.88916978153591398061446582156, −3.79268926481214292886516402263, −1.42904875784079428916030673770,
0.62131639840859696138665275066, 1.98280968963931956442013184348, 4.87407596767268453233700348715, 5.74984665983971706171155602659, 7.10360811679720904067814639335, 8.433742294964248988756110316453, 9.343945915499801769534803063436, 10.02732599381832542361248355163, 10.95491779956213013773742835883, 12.23197141272811162643667291006