| L(s) = 1 | + (0.781 − 0.623i)2-s + (0.558 + 1.63i)3-s + (0.222 − 0.974i)4-s + (0.306 − 4.09i)5-s + (1.45 + 0.933i)6-s + (0.725 − 2.54i)7-s + (−0.433 − 0.900i)8-s + (−2.37 + 1.83i)9-s + (−2.31 − 3.39i)10-s + (−2.96 − 1.16i)11-s + (1.72 − 0.179i)12-s + (2.03 + 0.798i)13-s + (−1.01 − 2.44i)14-s + (6.88 − 1.78i)15-s + (−0.900 − 0.433i)16-s + (−3.67 − 1.13i)17-s + ⋯ |
| L(s) = 1 | + (0.552 − 0.440i)2-s + (0.322 + 0.946i)3-s + (0.111 − 0.487i)4-s + (0.137 − 1.83i)5-s + (0.595 + 0.381i)6-s + (0.274 − 0.961i)7-s + (−0.153 − 0.318i)8-s + (−0.791 + 0.610i)9-s + (−0.731 − 1.07i)10-s + (−0.893 − 0.350i)11-s + (0.497 − 0.0519i)12-s + (0.563 + 0.221i)13-s + (−0.272 − 0.652i)14-s + (1.77 − 0.460i)15-s + (−0.225 − 0.108i)16-s + (−0.890 − 0.274i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.313 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.21533 - 1.68171i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21533 - 1.68171i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.558 - 1.63i)T \) |
| 7 | \( 1 + (-0.725 + 2.54i)T \) |
| good | 5 | \( 1 + (-0.306 + 4.09i)T + (-4.94 - 0.745i)T^{2} \) |
| 11 | \( 1 + (2.96 + 1.16i)T + (8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (-2.03 - 0.798i)T + (9.52 + 8.84i)T^{2} \) |
| 17 | \( 1 + (3.67 + 1.13i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (-2.94 + 1.70i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.10 - 6.57i)T + (-1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-1.09 + 3.54i)T + (-23.9 - 16.3i)T^{2} \) |
| 31 | \( 1 - 1.28iT - 31T^{2} \) |
| 37 | \( 1 + (-7.38 + 6.85i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (-6.81 - 4.64i)T + (14.9 + 38.1i)T^{2} \) |
| 43 | \( 1 + (-9.31 + 6.34i)T + (15.7 - 40.0i)T^{2} \) |
| 47 | \( 1 + (-5.92 - 7.42i)T + (-10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (-0.410 + 0.442i)T + (-3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (0.384 + 0.185i)T + (36.7 + 46.1i)T^{2} \) |
| 61 | \( 1 + (-4.94 + 1.12i)T + (54.9 - 26.4i)T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 + (-9.27 - 2.11i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-0.503 + 0.197i)T + (53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 + (1.01 + 2.59i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (3.25 + 0.490i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 + (2.91 + 1.68i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.647783172558014375297103995982, −9.391197882692671465018319785024, −8.298512128687438530578749535567, −7.66463370941393232384210743828, −5.85015953056733833882228815342, −5.23906194883272559754033147854, −4.31723245174487585637330947254, −3.90255823859386061931261224725, −2.30169602919219081691660117364, −0.78231220609636885033462399784,
2.36009795669075477106043481102, 2.62264480140627609413799524938, 3.89359135143819491325614173245, 5.51843016573087605708689357709, 6.25034919958628258778485682052, 6.78184517385554577703175114218, 7.81655024644103318740019614078, 8.226984409923120164515921083320, 9.481111378819328420479882936383, 10.63380465099841462898953576867
