| L(s) = 1 | + (0.781 + 0.623i)2-s + (−1.58 + 0.689i)3-s + (0.222 + 0.974i)4-s + (0.185 + 2.48i)5-s + (−1.67 − 0.451i)6-s + (−2.26 + 1.36i)7-s + (−0.433 + 0.900i)8-s + (2.04 − 2.19i)9-s + (−1.40 + 2.05i)10-s + (−1.02 + 0.402i)11-s + (−1.02 − 1.39i)12-s + (−1.42 + 0.559i)13-s + (−2.62 − 0.347i)14-s + (−2.00 − 3.81i)15-s + (−0.900 + 0.433i)16-s + (−2.55 + 0.786i)17-s + ⋯ |
| L(s) = 1 | + (0.552 + 0.440i)2-s + (−0.917 + 0.398i)3-s + (0.111 + 0.487i)4-s + (0.0831 + 1.10i)5-s + (−0.682 − 0.184i)6-s + (−0.856 + 0.515i)7-s + (−0.153 + 0.318i)8-s + (0.682 − 0.730i)9-s + (−0.443 + 0.649i)10-s + (−0.309 + 0.121i)11-s + (−0.296 − 0.402i)12-s + (−0.395 + 0.155i)13-s + (−0.700 − 0.0928i)14-s + (−0.517 − 0.984i)15-s + (−0.225 + 0.108i)16-s + (−0.618 + 0.190i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.730 + 0.682i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.730 + 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.252217 - 0.639178i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.252217 - 0.639178i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.781 - 0.623i)T \) |
| 3 | \( 1 + (1.58 - 0.689i)T \) |
| 7 | \( 1 + (2.26 - 1.36i)T \) |
| good | 5 | \( 1 + (-0.185 - 2.48i)T + (-4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (1.02 - 0.402i)T + (8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (1.42 - 0.559i)T + (9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (2.55 - 0.786i)T + (14.0 - 9.57i)T^{2} \) |
| 19 | \( 1 + (-2.02 - 1.16i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.570 + 0.614i)T + (-1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (2.29 + 7.43i)T + (-23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + 0.774iT - 31T^{2} \) |
| 37 | \( 1 + (-3.75 - 3.48i)T + (2.76 + 36.8i)T^{2} \) |
| 41 | \( 1 + (-1.87 + 1.27i)T + (14.9 - 38.1i)T^{2} \) |
| 43 | \( 1 + (6.51 + 4.44i)T + (15.7 + 40.0i)T^{2} \) |
| 47 | \( 1 + (1.59 - 1.99i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (-0.355 - 0.382i)T + (-3.96 + 52.8i)T^{2} \) |
| 59 | \( 1 + (12.2 - 5.88i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (7.81 + 1.78i)T + (54.9 + 26.4i)T^{2} \) |
| 67 | \( 1 + 1.03T + 67T^{2} \) |
| 71 | \( 1 + (7.60 - 1.73i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-14.6 - 5.76i)T + (53.5 + 49.6i)T^{2} \) |
| 79 | \( 1 + 3.42T + 79T^{2} \) |
| 83 | \( 1 + (5.32 - 13.5i)T + (-60.8 - 56.4i)T^{2} \) |
| 89 | \( 1 + (-7.61 + 1.14i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 + (0.936 - 0.540i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.67241334997183866695532902905, −9.943290078336336570601005428349, −9.201049639456143665008337781123, −7.80893904577493791454353113912, −6.86576058297861632782980804608, −6.30985499913474967423508140518, −5.61673858788896774253501109175, −4.52950718410865717340398077517, −3.49224496363064662811452749391, −2.46619563450844161835293658942,
0.30229485275445093613401687477, 1.55715808818629491031121619227, 3.11386964760962194449872627556, 4.46905243131140115978623043836, 5.06895323937070160063325958744, 5.96184718833396327104861510287, 6.85683213146797301121485305982, 7.71239386282615085041796032985, 9.018002424886362108133322061198, 9.756028025646719857872887092140
