L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (0.499 − 0.866i)6-s − 4i·7-s + 0.999i·8-s + (−1 + 1.73i)9-s + 3·11-s + 0.999i·12-s + (1.73 + i)13-s + (2 + 3.46i)14-s + (−0.5 − 0.866i)16-s + (−5.19 + 3i)17-s − 2i·18-s + (3.5 − 2.59i)19-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (0.204 − 0.353i)6-s − 1.51i·7-s + 0.353i·8-s + (−0.333 + 0.577i)9-s + 0.904·11-s + 0.288i·12-s + (0.480 + 0.277i)13-s + (0.534 + 0.925i)14-s + (−0.125 − 0.216i)16-s + (−1.26 + 0.727i)17-s − 0.471i·18-s + (0.802 − 0.596i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.466 + 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.634675 - 0.382663i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.634675 - 0.382663i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-3.5 + 2.59i)T \) |
good | 3 | \( 1 + (0.866 - 0.5i)T + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + (-1.73 - i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (5.19 - 3i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (5.19 + 3i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.46 + 2i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.19 - 3i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.5 + 7.79i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.06 - 3.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.866 + 0.5i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 3iT - 83T^{2} \) |
| 89 | \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-14.7 + 8.5i)T + (48.5 - 84.0i)T^{2} \) |
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
−9.999938917270649970287659278206, −9.036013554772395989147657355060, −8.258041408658967622893955828608, −7.27331410165023910476046279217, −6.62718342087125313502679838885, −5.74695227921424646434916344299, −4.51059847412809537770235779511, −3.85084854726474291964734602701, −1.99123612801223462201687630043, −0.49204898558125110267532481645,
1.29940838718436080897476112136, 2.58504170256476707877433006267, 3.64398279032656960535210406056, 5.10107888811314782736796660770, 6.13753774016493014430685776534, 6.56491162875177472362685968503, 7.86286925961538789347959929882, 8.745830232621326750504714274519, 9.253883584991822195300021768731, 10.02153335175736849314243994800