L(s) = 1 | + 2-s + (1.67 + 0.438i)3-s + 4-s + (1.57 + 0.908i)5-s + (1.67 + 0.438i)6-s + (−2.47 + 0.947i)7-s + 8-s + (2.61 + 1.46i)9-s + (1.57 + 0.908i)10-s + (−1.02 + 1.77i)11-s + (1.67 + 0.438i)12-s + (−3.57 + 0.463i)13-s + (−2.47 + 0.947i)14-s + (2.23 + 2.21i)15-s + 16-s + 5.68·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.967 + 0.253i)3-s + 0.5·4-s + (0.703 + 0.406i)5-s + (0.684 + 0.178i)6-s + (−0.933 + 0.358i)7-s + 0.353·8-s + (0.871 + 0.489i)9-s + (0.497 + 0.287i)10-s + (−0.309 + 0.535i)11-s + (0.483 + 0.126i)12-s + (−0.991 + 0.128i)13-s + (−0.660 + 0.253i)14-s + (0.577 + 0.571i)15-s + 0.250·16-s + 1.37·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.811 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.86370 + 0.922925i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.86370 + 0.922925i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (-1.67 - 0.438i)T \) |
| 7 | \( 1 + (2.47 - 0.947i)T \) |
| 13 | \( 1 + (3.57 - 0.463i)T \) |
good | 5 | \( 1 + (-1.57 - 0.908i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.02 - 1.77i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 - 5.68T + 17T^{2} \) |
| 19 | \( 1 + (0.796 + 1.38i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 8.73iT - 23T^{2} \) |
| 29 | \( 1 + (-0.724 + 0.418i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.97 + 6.88i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4.21iT - 37T^{2} \) |
| 41 | \( 1 + (-0.397 + 0.229i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.836 + 1.44i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.94 + 1.12i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.497 + 0.286i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 2.19iT - 59T^{2} \) |
| 61 | \( 1 + (5.97 - 3.44i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.46 + 2.57i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.24 + 7.34i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.11 - 8.86i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.68 - 6.38i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 15.0iT - 83T^{2} \) |
| 89 | \( 1 + 3.86iT - 89T^{2} \) |
| 97 | \( 1 + (1.72 - 2.98i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.58211564303303064551092570515, −9.956365562511176923636352781853, −9.412285768018712912477546861214, −8.146095394970072091289476076087, −7.17308952441813084253076368122, −6.32505221607361622002961690608, −5.20888712817169432896026077394, −4.10281878231057157399301560852, −2.84277863937365266174416274102, −2.28754318413548731401292433700,
1.56171617527279828577172972417, 2.99684868385960683853939616134, 3.66360583726351617129748091458, 5.17102364853822235434930872969, 5.99341819457373175284789309354, 7.21000246895694164007656658819, 7.79450806011310378290939161997, 9.144957097188330059459338281814, 9.749343916904733617625346467594, 10.49654572263595776891888683970