| L(s) = 1 | + (−0.873 − 1.11i)2-s + (−0.342 + 0.939i)3-s + (−0.474 + 1.94i)4-s + (−3.30 + 0.582i)5-s + (1.34 − 0.440i)6-s + (−1.96 + 3.40i)7-s + (2.57 − 1.16i)8-s + (−0.766 − 0.642i)9-s + (3.53 + 3.16i)10-s + (2.32 − 1.34i)11-s + (−1.66 − 1.11i)12-s + (−1.43 − 3.95i)13-s + (5.49 − 0.786i)14-s + (0.582 − 3.30i)15-s + (−3.55 − 1.84i)16-s + (4.40 − 3.69i)17-s + ⋯ |
| L(s) = 1 | + (−0.617 − 0.786i)2-s + (−0.197 + 0.542i)3-s + (−0.237 + 0.971i)4-s + (−1.47 + 0.260i)5-s + (0.548 − 0.179i)6-s + (−0.742 + 1.28i)7-s + (0.910 − 0.413i)8-s + (−0.255 − 0.214i)9-s + (1.11 + 1.00i)10-s + (0.701 − 0.404i)11-s + (−0.480 − 0.320i)12-s + (−0.399 − 1.09i)13-s + (1.46 − 0.210i)14-s + (0.150 − 0.853i)15-s + (−0.887 − 0.460i)16-s + (1.06 − 0.896i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.486 + 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.143955 - 0.244888i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.143955 - 0.244888i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.873 + 1.11i)T \) |
| 3 | \( 1 + (0.342 - 0.939i)T \) |
| 19 | \( 1 + (3.87 - 2.00i)T \) |
| good | 5 | \( 1 + (3.30 - 0.582i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (1.96 - 3.40i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.32 + 1.34i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.43 + 3.95i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-4.40 + 3.69i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.958 + 5.43i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (0.439 - 0.524i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.25 + 2.16i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 4.16iT - 37T^{2} \) |
| 41 | \( 1 + (10.3 + 3.77i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-12.0 + 2.11i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-4.55 - 3.82i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (9.75 + 1.72i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (5.47 + 6.53i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.92 - 0.340i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (2.73 - 3.26i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (1.84 + 10.4i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (13.9 + 5.06i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (3.82 + 1.39i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (1.58 + 0.912i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (1.91 - 0.698i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-8.85 + 7.43i)T + (16.8 - 95.5i)T^{2} \) |
| 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
−10.78079492476458916028542873698, −9.974920759655935948005858828499, −8.999312937388231977770306327982, −8.326510763535219707866779306223, −7.39845828780184679636498148781, −6.07915277008584850955872428029, −4.66302236201608538610423812077, −3.46783535219255115087122568141, −2.84448157526198865887716792938, −0.25237758222940949428560236107,
1.26010682385827581700797566862, 3.79984080149204861101063715014, 4.54487140083482409552193040326, 6.15017478095005629805509905116, 7.16501066280253987464613071071, 7.38654696234830829888196653514, 8.445422366103119153626423183233, 9.420700490367132967714434404209, 10.40423090359260693804169416669, 11.30690406157748767990226418615
