L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (2 + 3.46i)5-s + 0.999·6-s + (−0.5 + 2.59i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (1.99 − 3.46i)10-s + (0.5 − 0.866i)11-s + (−0.499 − 0.866i)12-s − 13-s + (2.5 − 0.866i)14-s − 3.99·15-s + (−0.5 − 0.866i)16-s + (−1.5 + 2.59i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.894 + 1.54i)5-s + 0.408·6-s + (−0.188 + 0.981i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.632 − 1.09i)10-s + (0.150 − 0.261i)11-s + (−0.144 − 0.249i)12-s − 0.277·13-s + (0.668 − 0.231i)14-s − 1.03·15-s + (−0.125 − 0.216i)16-s + (−0.363 + 0.630i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.615496 + 0.809059i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.615496 + 0.809059i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.5 - 2.59i)T \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 + (-2 - 3.46i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4 - 6.92i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 + (-5.5 - 9.52i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.5 + 4.33i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.5 - 12.9i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.5 + 4.33i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 15T + 71T^{2} \) |
| 73 | \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1 - 1.73i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−11.01473005648173930704497128215, −10.13100799836595541037195690828, −9.639608976800903749041328363820, −8.727483960414888247017924953855, −7.49314712194292955504532570063, −6.21544430431885191259355764153, −5.82952127220570794917127389005, −4.18288663790755686694712869021, −2.92148731815786704734291716126, −2.18116647876717199780574117014,
0.68084078626506192302658611439, 1.87619735147267196865161535985, 4.16373497743570463208721291599, 5.17638968322698390520217871157, 5.87840156796633950753714543275, 7.04173536243272232082857831152, 7.72388723505297507276321267099, 8.872462385976374323665888815038, 9.479161326981402472131254815057, 10.25481328194648589163187790571