L(s) = 1 | + (−0.942 − 1.05i)2-s + (0.438 − 1.63i)3-s + (−0.223 + 1.98i)4-s + (−0.615 + 2.14i)5-s + (−2.13 + 1.07i)6-s + (−0.885 + 2.49i)7-s + (2.30 − 1.63i)8-s + (0.114 + 0.0663i)9-s + (2.84 − 1.37i)10-s + (2.94 + 5.09i)11-s + (3.15 + 1.23i)12-s + (1.49 − 1.49i)13-s + (3.46 − 1.41i)14-s + (3.24 + 1.94i)15-s + (−3.90 − 0.888i)16-s + (−5.43 − 1.45i)17-s + ⋯ |
L(s) = 1 | + (−0.666 − 0.745i)2-s + (0.253 − 0.944i)3-s + (−0.111 + 0.993i)4-s + (−0.275 + 0.961i)5-s + (−0.872 + 0.440i)6-s + (−0.334 + 0.942i)7-s + (0.815 − 0.578i)8-s + (0.0383 + 0.0221i)9-s + (0.900 − 0.435i)10-s + (0.887 + 1.53i)11-s + (0.910 + 0.356i)12-s + (0.414 − 0.414i)13-s + (0.925 − 0.378i)14-s + (0.838 + 0.503i)15-s + (−0.975 − 0.222i)16-s + (−1.31 − 0.353i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0305i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.922457 + 0.0140981i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.922457 + 0.0140981i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.942 + 1.05i)T \) |
| 5 | \( 1 + (0.615 - 2.14i)T \) |
| 7 | \( 1 + (0.885 - 2.49i)T \) |
good | 3 | \( 1 + (-0.438 + 1.63i)T + (-2.59 - 1.5i)T^{2} \) |
| 11 | \( 1 + (-2.94 - 5.09i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.49 + 1.49i)T - 13iT^{2} \) |
| 17 | \( 1 + (5.43 + 1.45i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-3.10 - 1.79i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.669 - 2.49i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 6.02T + 29T^{2} \) |
| 31 | \( 1 + (-1.05 + 0.609i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-10.0 + 2.68i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 2.44T + 41T^{2} \) |
| 43 | \( 1 + (-7.70 - 7.70i)T + 43iT^{2} \) |
| 47 | \( 1 + (4.15 - 1.11i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (4.68 + 1.25i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-2.51 + 1.45i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.34 - 3.66i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (8.27 + 2.21i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 2.68iT - 71T^{2} \) |
| 73 | \( 1 + (-0.400 + 1.49i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-1.03 + 1.78i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.700 + 0.700i)T + 83iT^{2} \) |
| 89 | \( 1 + (3.21 + 1.85i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.71 + 2.71i)T - 97iT^{2} \) |
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
−11.78019453347314684105559642259, −11.12852673835052127208201912519, −9.814827479193164464562737563739, −9.218356673571249033783879836208, −7.87558298089095110643837554062, −7.23642428325418609945532806405, −6.33531221211754084826436066026, −4.23557588589277677465555928959, −2.78637844522795517988085096119, −1.82486321722199469349788890691,
0.927688389244265385437738562220, 3.81838238155826385098345550388, 4.54812342277155048573587510654, 5.96208998832051717623157820618, 6.99038647642239884150777186063, 8.293932646386039121642231217204, 9.070847348920012419072653600966, 9.542801067271453026598494773775, 10.81197654960611264753711993537, 11.37376012297603350303566160340