L(s) = 1 | + (2.19 − 0.588i)3-s + (−1.19 − 1.88i)5-s + (1.40 + 2.24i)7-s + (1.88 − 1.08i)9-s + (1.78 − 3.09i)11-s + (3.13 + 3.13i)13-s + (−3.74 − 3.44i)15-s + (−1.34 − 5.00i)17-s + (−0.687 − 1.19i)19-s + (4.40 + 4.10i)21-s + (−4.00 − 1.07i)23-s + (−2.12 + 4.52i)25-s + (−1.32 + 1.32i)27-s + 9.39i·29-s + (−6.08 − 3.51i)31-s + ⋯ |
L(s) = 1 | + (1.26 − 0.340i)3-s + (−0.536 − 0.844i)5-s + (0.530 + 0.847i)7-s + (0.628 − 0.363i)9-s + (0.539 − 0.934i)11-s + (0.869 + 0.869i)13-s + (−0.967 − 0.888i)15-s + (−0.325 − 1.21i)17-s + (−0.157 − 0.273i)19-s + (0.961 + 0.895i)21-s + (−0.835 − 0.223i)23-s + (−0.425 + 0.905i)25-s + (−0.254 + 0.254i)27-s + 1.74i·29-s + (−1.09 − 0.630i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.523i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.852 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.74642 - 0.493330i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.74642 - 0.493330i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.19 + 1.88i)T \) |
| 7 | \( 1 + (-1.40 - 2.24i)T \) |
good | 3 | \( 1 + (-2.19 + 0.588i)T + (2.59 - 1.5i)T^{2} \) |
| 11 | \( 1 + (-1.78 + 3.09i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.13 - 3.13i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.34 + 5.00i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (0.687 + 1.19i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.00 + 1.07i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 - 9.39iT - 29T^{2} \) |
| 31 | \( 1 + (6.08 + 3.51i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.68 - 6.29i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 4.63iT - 41T^{2} \) |
| 43 | \( 1 + (-2.51 + 2.51i)T - 43iT^{2} \) |
| 47 | \( 1 + (-8.76 - 2.34i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (0.470 + 1.75i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (1.42 - 2.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.91 + 1.10i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.37 - 1.17i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 + (-2.44 + 0.654i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (4.82 - 2.78i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.863 - 0.863i)T + 83iT^{2} \) |
| 89 | \( 1 + (-0.430 - 0.745i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-12.6 + 12.6i)T - 97iT^{2} \) |
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
−11.75913037313035329417652903986, −11.16345112802570445899262945248, −9.221430157466288341557696055161, −8.865432754656495232754240751820, −8.254395358303090858229153348326, −7.14125116601134650815903997046, −5.68981845416977532181853682452, −4.34832812303559429032388258796, −3.12643957805687919002386385980, −1.64529608170274148571938404233,
2.10163300573331531316442339363, 3.73152143627886939848096279671, 4.05809328334975758667296726279, 6.10392840583362738545893757811, 7.44840334892032758916534475031, 7.986639738555875973116082489107, 8.968876544388731073867756977727, 10.21134568897799897984496122826, 10.70339270973247993767968796610, 11.87054888214857169050899214441