L(s) = 1 | + (−1.38 + 0.305i)2-s + (1.81 − 0.844i)4-s − 5-s + 1.41i·7-s + (−2.24 + 1.71i)8-s + (1.38 − 0.305i)10-s − 0.191i·11-s + 2.63i·13-s + (−0.432 − 1.95i)14-s + (2.57 − 3.06i)16-s + 6.20i·17-s − 1.52·19-s + (−1.81 + 0.844i)20-s + (0.0585 + 0.264i)22-s + 5.25·23-s + ⋯ |
L(s) = 1 | + (−0.976 + 0.216i)2-s + (0.906 − 0.422i)4-s − 0.447·5-s + 0.534i·7-s + (−0.793 + 0.608i)8-s + (0.436 − 0.0966i)10-s − 0.0577i·11-s + 0.731i·13-s + (−0.115 − 0.521i)14-s + (0.643 − 0.765i)16-s + 1.50i·17-s − 0.349·19-s + (−0.405 + 0.188i)20-s + (0.0124 + 0.0563i)22-s + 1.09·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0381 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0381 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.471545 + 0.489891i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.471545 + 0.489891i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.38 - 0.305i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
good | 7 | \( 1 - 1.41iT - 7T^{2} \) |
| 11 | \( 1 + 0.191iT - 11T^{2} \) |
| 13 | \( 1 - 2.63iT - 13T^{2} \) |
| 17 | \( 1 - 6.20iT - 17T^{2} \) |
| 19 | \( 1 + 1.52T + 19T^{2} \) |
| 23 | \( 1 - 5.25T + 23T^{2} \) |
| 29 | \( 1 - 0.270T + 29T^{2} \) |
| 31 | \( 1 - 6.20iT - 31T^{2} \) |
| 37 | \( 1 - 7.61iT - 37T^{2} \) |
| 41 | \( 1 - 9.22iT - 41T^{2} \) |
| 43 | \( 1 + 12.7T + 43T^{2} \) |
| 47 | \( 1 + 3.79T + 47T^{2} \) |
| 53 | \( 1 - 8.77T + 53T^{2} \) |
| 59 | \( 1 + 10.4iT - 59T^{2} \) |
| 61 | \( 1 + 0.382iT - 61T^{2} \) |
| 67 | \( 1 - 1.72T + 67T^{2} \) |
| 71 | \( 1 - 9.72T + 71T^{2} \) |
| 73 | \( 1 + 5.45T + 73T^{2} \) |
| 79 | \( 1 + 14.3iT - 79T^{2} \) |
| 83 | \( 1 + 15.2iT - 83T^{2} \) |
| 89 | \( 1 - 3.56iT - 89T^{2} \) |
| 97 | \( 1 - 7.31T + 97T^{2} \) |
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
−11.51646526526080049744770709027, −10.68153700578713028179883497690, −9.775487979158851297932688040122, −8.670402631112199777672128820850, −8.262869201625279357297394016928, −6.95827975598520280270052803775, −6.23514234043085971187256959208, −4.89351865727131501774244785814, −3.23740396396156240183684737019, −1.66626809346835669928442269973,
0.65039299344575439674815142797, 2.59070308067302866349665216807, 3.83524111825846302276724745974, 5.37638573831189931998924952720, 6.89425252288544559253851652305, 7.45499721792535724660743020827, 8.466710833981547157151772490688, 9.373286685741437464177548154427, 10.29858573375641786436050351359, 11.08465963664232259597760521013