L(s) = 1 | + (0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (−1.37 + 1.76i)5-s + (0.499 + 0.866i)6-s + 0.785i·7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−2.07 + 0.837i)10-s − 0.377·11-s + 0.999i·12-s + (−2.51 + 1.45i)13-s + (−0.392 + 0.680i)14-s + (−2.07 + 0.837i)15-s + (−0.5 + 0.866i)16-s + (2.45 + 1.41i)17-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.615 + 0.788i)5-s + (0.204 + 0.353i)6-s + 0.296i·7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.655 + 0.264i)10-s − 0.113·11-s + 0.288i·12-s + (−0.697 + 0.402i)13-s + (−0.104 + 0.181i)14-s + (−0.535 + 0.216i)15-s + (−0.125 + 0.216i)16-s + (0.595 + 0.343i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.404 - 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.404 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.10993 + 1.70459i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.10993 + 1.70459i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 + (1.37 - 1.76i)T \) |
| 19 | \( 1 + (2.82 - 3.32i)T \) |
good | 7 | \( 1 - 0.785iT - 7T^{2} \) |
| 11 | \( 1 + 0.377T + 11T^{2} \) |
| 13 | \( 1 + (2.51 - 1.45i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.45 - 1.41i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-7.86 + 4.54i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.01 - 3.48i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 4.42T + 31T^{2} \) |
| 37 | \( 1 + 0.0967iT - 37T^{2} \) |
| 41 | \( 1 + (1.43 - 2.48i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.371 + 0.214i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-10.4 + 6.04i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.00 + 3.46i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.88 - 8.45i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.27 + 3.94i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.31 + 4.22i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.86 + 10.1i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (1.95 + 1.13i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.785 - 1.36i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 2.75iT - 83T^{2} \) |
| 89 | \( 1 + (-5.96 - 10.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.98 - 1.72i)T + (48.5 + 84.0i)T^{2} \) |
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
−10.91134724459664737624750643702, −10.35737635180636625059234910585, −9.096343993224343555549570002310, −8.250107839295989566212449728512, −7.33410076475983215452191050852, −6.62899170632847752275342114075, −5.39155062866898386048797728357, −4.31799581864338685586627577869, −3.37003789005943331196079045226, −2.37640281161383854380499734481,
0.936612609738759809173212533708, 2.58219799097758715240057134992, 3.69393050090537149817444791315, 4.72032299675113300795464002183, 5.56297679894708600013864996071, 7.07749438200636669819255718100, 7.62881545949417393491327492747, 8.786474947030077203458393565939, 9.500083203290322590186338224300, 10.61759043371806523783989109791