L(s) = 1 | + (−1.57 − 0.907i)2-s + (0.645 + 1.11i)4-s + (1.19 − 1.88i)5-s + (−4.06 + 2.34i)7-s + 1.28i·8-s + (−3.59 + 1.88i)10-s + (0.270 − 0.468i)11-s + (1.34 + 3.34i)13-s + 8.50·14-s + (2.45 − 4.25i)16-s + (5.92 − 3.42i)17-s + (2.78 + 4.81i)19-s + (2.88 + 0.118i)20-s + (−0.849 + 0.490i)22-s + (−0.0291 − 0.0168i)23-s + ⋯ |
L(s) = 1 | + (−1.11 − 0.641i)2-s + (0.322 + 0.559i)4-s + (0.535 − 0.844i)5-s + (−1.53 + 0.886i)7-s + 0.454i·8-s + (−1.13 + 0.595i)10-s + (0.0815 − 0.141i)11-s + (0.372 + 0.928i)13-s + 2.27·14-s + (0.614 − 1.06i)16-s + (1.43 − 0.829i)17-s + (0.638 + 1.10i)19-s + (0.645 + 0.0266i)20-s + (−0.181 + 0.104i)22-s + (−0.00607 − 0.00350i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.715631 - 0.164362i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.715631 - 0.164362i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1.19 + 1.88i)T \) |
| 13 | \( 1 + (-1.34 - 3.34i)T \) |
good | 2 | \( 1 + (1.57 + 0.907i)T + (1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (4.06 - 2.34i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.270 + 0.468i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-5.92 + 3.42i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.78 - 4.81i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.0291 + 0.0168i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.40 - 5.90i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 0.352T + 31T^{2} \) |
| 37 | \( 1 + (-6.98 - 4.03i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.59 + 6.23i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-6.08 + 3.51i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 1.99iT - 47T^{2} \) |
| 53 | \( 1 + 8.50iT - 53T^{2} \) |
| 59 | \( 1 + (-6.64 - 11.5i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.60 - 2.77i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.70 - 5.02i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.33 + 7.50i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 8.05iT - 73T^{2} \) |
| 79 | \( 1 + 2.29T + 79T^{2} \) |
| 83 | \( 1 - 14.8iT - 83T^{2} \) |
| 89 | \( 1 + (-3.98 + 6.90i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.265 - 0.153i)T + (48.5 - 84.0i)T^{2} \) |
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
−10.16943781283446840328418359742, −9.689401155522103827241367190420, −9.140369709884017720681701060121, −8.489737010875160920036442653901, −7.26982825781569315069077080503, −5.92214681091087103303969607474, −5.37895039910580479501796975874, −3.57198826921281673233738251463, −2.38434422560148450877356513669, −1.04285818847827754010129334630,
0.790476381100226881610391822841, 2.98681884392256010844333887129, 3.82335770897721601348537513978, 5.89065831269803458345199758296, 6.37122067929122241080116190873, 7.40380165279256062159030474552, 7.81764096873307737010770124550, 9.302292412943489847943294468215, 9.768403626192942270099885855016, 10.33396889872090138772218109523