L(s) = 1 | + (−0.766 + 0.642i)2-s + (0.326 + 0.118i)3-s + (0.173 − 0.984i)4-s + (−0.326 + 0.118i)6-s + (0.500 + 0.866i)8-s + (−0.673 − 0.565i)9-s + (−0.766 − 1.32i)11-s + (0.173 − 0.300i)12-s + (−0.939 − 0.342i)16-s + (−0.766 + 0.642i)17-s + 0.879·18-s + (1.43 + 0.524i)22-s + (0.0603 + 0.342i)24-s + (−0.939 + 0.342i)25-s + (−0.326 − 0.565i)27-s + ⋯ |
L(s) = 1 | + (−0.766 + 0.642i)2-s + (0.326 + 0.118i)3-s + (0.173 − 0.984i)4-s + (−0.326 + 0.118i)6-s + (0.500 + 0.866i)8-s + (−0.673 − 0.565i)9-s + (−0.766 − 1.32i)11-s + (0.173 − 0.300i)12-s + (−0.939 − 0.342i)16-s + (−0.766 + 0.642i)17-s + 0.879·18-s + (1.43 + 0.524i)22-s + (0.0603 + 0.342i)24-s + (−0.939 + 0.342i)25-s + (−0.326 − 0.565i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.363 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.363 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3264967351\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3264967351\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-0.326 - 0.118i)T + (0.766 + 0.642i)T^{2} \) |
| 5 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 17 | \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 23 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 29 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (1.76 + 0.642i)T + (0.766 + 0.642i)T^{2} \) |
| 43 | \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 47 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 53 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 59 | \( 1 + (1.17 - 0.984i)T + (0.173 - 0.984i)T^{2} \) |
| 61 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 67 | \( 1 + (-1.43 - 1.20i)T + (0.173 + 0.984i)T^{2} \) |
| 71 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 73 | \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \) |
| 79 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 83 | \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 97 | \( 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2} \) |
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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
−8.586710958407237614858675650121, −8.244441776892953788713582975148, −7.33339325268218174547959317123, −6.42949714246979090534347078982, −5.81895619500669336976867520901, −5.17195224370701747702947842696, −3.89263359034411314783900896050, −2.95986115928077477386995662327, −1.85088629190448296435541598871, −0.23125042384434741832610571197,
1.77459382425977294508705141417, 2.44304109337064591319266065701, 3.25559517346108979056305973019, 4.46389024739387989570389346269, 5.09471918037778321953593550912, 6.41212603316914391826751411886, 7.20548987273692779338045741083, 7.949461419968160632042873099519, 8.313668722892315862808242573043, 9.378623351359094034659401414613