L(s) = 1 | + (−1.64 + 0.951i)2-s + (−0.866 − 0.5i)3-s + (0.811 − 1.40i)4-s + (−2.07 + 0.837i)5-s + 1.90·6-s − 0.719i·8-s + (0.499 + 0.866i)9-s + (2.61 − 3.35i)10-s + (−1 + 1.73i)11-s + (−1.40 + 0.811i)12-s + 6.42i·13-s + (2.21 + 0.311i)15-s + (2.30 + 3.99i)16-s + (−3.83 − 2.21i)17-s + (−1.64 − 0.951i)18-s + (−1.21 − 2.10i)19-s + ⋯ |
L(s) = 1 | + (−1.16 + 0.672i)2-s + (−0.499 − 0.288i)3-s + (0.405 − 0.702i)4-s + (−0.927 + 0.374i)5-s + 0.776·6-s − 0.254i·8-s + (0.166 + 0.288i)9-s + (0.828 − 1.06i)10-s + (−0.301 + 0.522i)11-s + (−0.405 + 0.234i)12-s + 1.78i·13-s + (0.571 + 0.0803i)15-s + (0.576 + 0.998i)16-s + (−0.930 − 0.537i)17-s + (−0.388 − 0.224i)18-s + (−0.278 − 0.482i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.146693 - 0.0859531i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.146693 - 0.0859531i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 + (2.07 - 0.837i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (1.64 - 0.951i)T + (1 - 1.73i)T^{2} \) |
| 11 | \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 6.42iT - 13T^{2} \) |
| 17 | \( 1 + (3.83 + 2.21i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.21 + 2.10i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.19 - 0.688i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 0.755T + 29T^{2} \) |
| 31 | \( 1 + (2.59 - 4.48i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.59 + 3.80i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 8.23T + 41T^{2} \) |
| 43 | \( 1 + 10.1iT - 43T^{2} \) |
| 47 | \( 1 + (-2.38 + 1.37i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-7.95 - 4.59i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-7.05 + 12.2i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.42 + 5.93i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.38 + 1.37i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + (1.36 + 0.785i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.42 + 4.20i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 11.6iT - 83T^{2} \) |
| 89 | \( 1 + (-2.31 - 4.00i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 11.9iT - 97T^{2} \) |
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
−10.13656000043652073786373537330, −9.121912527564780364901321279571, −8.569213336815676057107102113970, −7.39679183682057566478851615966, −7.04495957558962106882241431651, −6.36096129574354636464979919120, −4.81075889294926496032227060473, −3.87217821571288034782005773224, −2.03140870983165599720888098168, −0.17798544785847505235991490465,
0.968926579001728051071307376238, 2.72397167590472067478437314646, 3.91117165030837797596986586224, 5.11781897673331644958054161310, 6.02985204912436685690122262977, 7.51717433755350592180641590878, 8.220438530785552672056623236754, 8.741758531487030270631925181277, 9.891793436942263432759451553863, 10.50951854128333525374281413595