L(s) = 1 | + 3-s + (−2 − i)5-s − 4i·7-s + 9-s + 4i·11-s + (−2 − i)15-s + 4i·17-s − 4i·21-s + 4i·23-s + (3 + 4i)25-s + 27-s + 6i·29-s + 4·31-s + 4i·33-s + (−4 + 8i)35-s + ⋯ |
L(s) = 1 | + 0.577·3-s + (−0.894 − 0.447i)5-s − 1.51i·7-s + 0.333·9-s + 1.20i·11-s + (−0.516 − 0.258i)15-s + 0.970i·17-s − 0.872i·21-s + 0.834i·23-s + (0.600 + 0.800i)25-s + 0.192·27-s + 1.11i·29-s + 0.718·31-s + 0.696i·33-s + (−0.676 + 1.35i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.774280956\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.774280956\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + (2 + i)T \) |
good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 12T + 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 - 2iT - 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−8.438889038947173182413323343330, −7.68141460742279706607448359768, −7.25741095001970878006871203103, −6.67788901736851606326597663012, −5.30929832302428030216489175673, −4.47135175044073417687056955176, −3.94434004355725885132051536689, −3.31941608057261962450531382365, −1.89069860782359118672668043211, −0.955018373414104480512377092868,
0.59737789044940422032016358078, 2.34943906245832997555044402730, 2.83003538722097492246331937631, 3.63925703361201978622807868801, 4.59340973585300201209166650320, 5.51051806295697786845614915780, 6.26345118242283070785353508383, 7.02004650485077925950007305841, 7.965815070039479399631035195020, 8.426220001072337156911335852935