L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−2.19 + 0.448i)5-s − 0.999i·8-s + (−1.5 − 2.59i)9-s + (−1.67 + 1.48i)10-s + (2 − 3.46i)11-s − 4.24i·13-s + (−0.5 − 0.866i)16-s + (3.67 + 2.12i)17-s + (−2.59 − 1.5i)18-s + (−2.82 − 4.89i)19-s + (−0.707 + 2.12i)20-s − 3.99i·22-s + (4.59 − 1.96i)25-s + (−2.12 − 3.67i)26-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.979 + 0.200i)5-s − 0.353i·8-s + (−0.5 − 0.866i)9-s + (−0.529 + 0.469i)10-s + (0.603 − 1.04i)11-s − 1.17i·13-s + (−0.125 − 0.216i)16-s + (0.891 + 0.514i)17-s + (−0.612 − 0.353i)18-s + (−0.648 − 1.12i)19-s + (−0.158 + 0.474i)20-s − 0.852i·22-s + (0.919 − 0.392i)25-s + (−0.416 − 0.720i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.246 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.923618 - 1.18801i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.923618 - 1.18801i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 + (2.19 - 0.448i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 4.24iT - 13T^{2} \) |
| 17 | \( 1 + (-3.67 - 2.12i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.82 + 4.89i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + (2.82 - 4.89i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.19 - 3i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 1.41T + 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (10.3 + 6i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.65 + 9.79i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.53 - 6.12i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-10.3 - 6i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + (3.67 + 2.12i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 16.9iT - 83T^{2} \) |
| 89 | \( 1 + (-2.12 - 3.67i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 4.24iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.03853750636298828488943121088, −10.04477000678638452993412988392, −8.775091753594705387054698742605, −8.119936144651313031012720037258, −6.81089707745841915736830574382, −6.02303749702719412340202545865, −4.86170995054942863367183419711, −3.52800212076404499604615505075, −3.09704937886795151950199254526, −0.77000627600658959443498098976,
2.06242027240983658224898531033, 3.68806037444316483404578936004, 4.46843895622302966768504186026, 5.43893793763337942888231757605, 6.72782025922558308844980564944, 7.51141168466346644861524794384, 8.297909455311541037052044432675, 9.300992755405197279220301286038, 10.50014376298182370696465388736, 11.50671529557614710800130365784