L(s) = 1 | + (−0.5 + 0.866i)2-s + (1.71 + 0.243i)3-s + (−0.499 − 0.866i)4-s + (2.87 − 1.66i)5-s + (−1.06 + 1.36i)6-s + (−0.187 − 2.63i)7-s + 0.999·8-s + (2.88 + 0.834i)9-s + 3.32i·10-s − 1.48·11-s + (−0.646 − 1.60i)12-s + (−1.88 + 3.07i)13-s + (2.37 + 1.15i)14-s + (5.34 − 2.15i)15-s + (−0.5 + 0.866i)16-s + (−2.81 − 4.88i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.990 + 0.140i)3-s + (−0.249 − 0.433i)4-s + (1.28 − 0.743i)5-s + (−0.436 + 0.556i)6-s + (−0.0707 − 0.997i)7-s + 0.353·8-s + (0.960 + 0.278i)9-s + 1.05i·10-s − 0.447·11-s + (−0.186 − 0.463i)12-s + (−0.523 + 0.851i)13-s + (0.635 + 0.309i)14-s + (1.37 − 0.555i)15-s + (−0.125 + 0.216i)16-s + (−0.683 − 1.18i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0158i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0158i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.96035 + 0.0155191i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.96035 + 0.0155191i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-1.71 - 0.243i)T \) |
| 7 | \( 1 + (0.187 + 2.63i)T \) |
| 13 | \( 1 + (1.88 - 3.07i)T \) |
good | 5 | \( 1 + (-2.87 + 1.66i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + 1.48T + 11T^{2} \) |
| 17 | \( 1 + (2.81 + 4.88i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 - 5.36T + 19T^{2} \) |
| 23 | \( 1 + (3.47 + 2.00i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.127 + 0.0736i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.689 - 1.19i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-8.80 - 5.08i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.728 - 0.420i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.56 - 7.90i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (8.41 - 4.85i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-10.6 - 6.15i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.131 - 0.0757i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + 7.30iT - 61T^{2} \) |
| 67 | \( 1 - 9.94iT - 67T^{2} \) |
| 71 | \( 1 + (2.25 - 3.90i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (1.99 - 3.44i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.75 - 3.03i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 11.3iT - 83T^{2} \) |
| 89 | \( 1 + (-1.49 - 0.863i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.27 - 12.5i)T + (-48.5 - 84.0i)T^{2} \) |
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
−10.22202758567152170030018687132, −9.658346534265922889599911285356, −9.234351992782190385161530882744, −8.152061485035593830782292172594, −7.31301006972220205093251110613, −6.45298621091037780229412741885, −5.07897828693442288433375015703, −4.40105130308096651742102427964, −2.67105378453262013652431285474, −1.33532169333596677403361886210,
1.93105711966556339950242863231, 2.54737842124222657125950652359, 3.53954539634336364449507745733, 5.27494087292972020151753726119, 6.23279847257289965781382761632, 7.44000676420160450528887232166, 8.330289871136308559575532369334, 9.210116843548647888036623506085, 9.948525421926989188233363507186, 10.39090830038210157980118823348