L(s) = 1 | + (−1.80 + 1.04i)3-s + (−0.410 + 0.236i)5-s − 3.88·7-s + (0.675 − 1.17i)9-s + 6.19i·11-s + (4.97 + 2.87i)13-s + (0.494 − 0.856i)15-s + (0.442 + 0.766i)17-s + (4.19 − 1.19i)19-s + (7.01 − 4.04i)21-s + (−0.917 + 1.58i)23-s + (−2.38 + 4.13i)25-s − 3.43i·27-s + (−6.58 − 3.79i)29-s − 6.70·31-s + ⋯ |
L(s) = 1 | + (−1.04 + 0.602i)3-s + (−0.183 + 0.105i)5-s − 1.46·7-s + (0.225 − 0.390i)9-s + 1.86i·11-s + (1.38 + 0.797i)13-s + (0.127 − 0.221i)15-s + (0.107 + 0.185i)17-s + (0.961 − 0.274i)19-s + (1.53 − 0.883i)21-s + (−0.191 + 0.331i)23-s + (−0.477 + 0.827i)25-s − 0.661i·27-s + (−1.22 − 0.705i)29-s − 1.20·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2260667557\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2260667557\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-4.19 + 1.19i)T \) |
good | 3 | \( 1 + (1.80 - 1.04i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (0.410 - 0.236i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 3.88T + 7T^{2} \) |
| 11 | \( 1 - 6.19iT - 11T^{2} \) |
| 13 | \( 1 + (-4.97 - 2.87i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.442 - 0.766i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (0.917 - 1.58i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (6.58 + 3.79i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 6.70T + 31T^{2} \) |
| 37 | \( 1 + 1.54iT - 37T^{2} \) |
| 41 | \( 1 + (3.44 + 5.96i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (6.45 - 3.72i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-6.71 + 11.6i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.06 - 1.19i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (10.1 - 5.88i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.12 - 1.22i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.74 - 5.04i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.98 + 6.89i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (3.01 + 5.22i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.52 + 4.36i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 10.3iT - 83T^{2} \) |
| 89 | \( 1 + (1.84 - 3.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.213 + 0.369i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.14009577676864575306482236929, −9.623350496942387404172146162413, −8.988602807826390334644657629215, −7.46473028129161998281899969811, −6.90461976585277841838210167162, −5.97152289670517440814493306726, −5.31433918312280420603681937752, −4.12543811274295623126316359859, −3.54907883559732522780167313875, −1.88219687898259032295229213789,
0.12985828564977278660596589226, 1.09544008456051755873350112503, 3.22563532855132568011609672387, 3.57368422292767722517404593643, 5.41174755015018353689122576703, 5.99548955843686254599990737275, 6.37106227015260246515728540338, 7.43670964286751721353054997370, 8.412274908705067098759578813093, 9.163461757188415863633263415883