L(s) = 1 | + (−1.15 + 1.15i)3-s − 4.31i·7-s + 0.316i·9-s + (0.158 + 0.158i)11-s + (−2.31 + 2.31i)13-s + 5.31·17-s + (−3.15 + 3.15i)19-s + (5 + 5i)21-s − 6.31i·23-s + (−3.84 − 3.84i)27-s + (2 − 2i)29-s − 4.31·31-s − 0.366·33-s + (−7.31 − 7.31i)37-s − 5.36i·39-s + ⋯ |
L(s) = 1 | + (−0.668 + 0.668i)3-s − 1.63i·7-s + 0.105i·9-s + (0.0477 + 0.0477i)11-s + (−0.642 + 0.642i)13-s + 1.28·17-s + (−0.724 + 0.724i)19-s + (1.09 + 1.09i)21-s − 1.31i·23-s + (−0.739 − 0.739i)27-s + (0.371 − 0.371i)29-s − 0.775·31-s − 0.0638·33-s + (−1.20 − 1.20i)37-s − 0.859i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5529329032\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5529329032\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (1.15 - 1.15i)T - 3iT^{2} \) |
| 7 | \( 1 + 4.31iT - 7T^{2} \) |
| 11 | \( 1 + (-0.158 - 0.158i)T + 11iT^{2} \) |
| 13 | \( 1 + (2.31 - 2.31i)T - 13iT^{2} \) |
| 17 | \( 1 - 5.31T + 17T^{2} \) |
| 19 | \( 1 + (3.15 - 3.15i)T - 19iT^{2} \) |
| 23 | \( 1 + 6.31iT - 23T^{2} \) |
| 29 | \( 1 + (-2 + 2i)T - 29iT^{2} \) |
| 31 | \( 1 + 4.31T + 31T^{2} \) |
| 37 | \( 1 + (7.31 + 7.31i)T + 37iT^{2} \) |
| 41 | \( 1 - 5iT - 41T^{2} \) |
| 43 | \( 1 + (-5.63 - 5.63i)T + 43iT^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + (3.31 + 3.31i)T + 53iT^{2} \) |
| 59 | \( 1 + (5.31 + 5.31i)T + 59iT^{2} \) |
| 61 | \( 1 + (3.63 - 3.63i)T - 61iT^{2} \) |
| 67 | \( 1 + (-5.84 + 5.84i)T - 67iT^{2} \) |
| 71 | \( 1 + 4.63iT - 71T^{2} \) |
| 73 | \( 1 + 13.3iT - 73T^{2} \) |
| 79 | \( 1 - 2.31T + 79T^{2} \) |
| 83 | \( 1 + (3.84 - 3.84i)T - 83iT^{2} \) |
| 89 | \( 1 + 15.9iT - 89T^{2} \) |
| 97 | \( 1 + 6.63T + 97T^{2} \) |
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
−9.517305914600869558098185687151, −8.194671208009006106953799993617, −7.56784799465919307634905567825, −6.72888277825199960460731107870, −5.86500938545839904998731410339, −4.75807552414826604621545572745, −4.32083563492569149540620654939, −3.38044073543917522641275134886, −1.77647859965190234964265579580, −0.23776296625395420844673892244,
1.39499189875972813489482397856, 2.58043730290178397282234446142, 3.53806230497802213461465621626, 5.25306438508092288790433627524, 5.43986152941148180650489919020, 6.35770410954438566177540807864, 7.17300817005242356240627031167, 8.018530241923177283402558592950, 8.909679537980250350507081785897, 9.522533786161626309058014342452