| L(s) = 1 | + (0.439 + 0.487i)2-s + (1.74 + 0.777i)3-s + (0.163 − 1.56i)4-s + (3.49 + 0.742i)5-s + (0.388 + 1.19i)6-s + (−0.295 − 2.62i)7-s + (1.89 − 1.37i)8-s + (0.439 + 0.487i)9-s + (1.17 + 2.02i)10-s + (1.49 − 2.59i)12-s + (−1.82 + 5.62i)13-s + (1.15 − 1.29i)14-s + (5.52 + 4.01i)15-s + (−1.56 − 0.332i)16-s + (−1.10 + 1.23i)17-s + (−0.0450 + 0.428i)18-s + ⋯ |
| L(s) = 1 | + (0.310 + 0.345i)2-s + (1.00 + 0.449i)3-s + (0.0819 − 0.780i)4-s + (1.56 + 0.331i)5-s + (0.158 + 0.487i)6-s + (−0.111 − 0.993i)7-s + (0.670 − 0.486i)8-s + (0.146 + 0.162i)9-s + (0.370 + 0.641i)10-s + (0.433 − 0.749i)12-s + (−0.506 + 1.55i)13-s + (0.308 − 0.347i)14-s + (1.42 + 1.03i)15-s + (−0.391 − 0.0831i)16-s + (−0.268 + 0.298i)17-s + (−0.0106 + 0.101i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.127i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.19953 + 0.204612i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.19953 + 0.204612i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (0.295 + 2.62i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (-0.439 - 0.487i)T + (-0.209 + 1.98i)T^{2} \) |
| 3 | \( 1 + (-1.74 - 0.777i)T + (2.00 + 2.22i)T^{2} \) |
| 5 | \( 1 + (-3.49 - 0.742i)T + (4.56 + 2.03i)T^{2} \) |
| 13 | \( 1 + (1.82 - 5.62i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (1.10 - 1.23i)T + (-1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (0.154 + 1.47i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (1.67 - 2.89i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.49 + 1.81i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-6.92 + 1.47i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (4.12 - 1.83i)T + (24.7 - 27.4i)T^{2} \) |
| 41 | \( 1 + (-1.04 + 0.755i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 1.59T + 43T^{2} \) |
| 47 | \( 1 + (-0.173 - 1.64i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (9.02 - 1.91i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (0.925 - 8.80i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + (-6.54 - 1.39i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-4.91 - 8.50i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.66 + 8.19i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (0.476 - 4.53i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (4.27 + 4.75i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (-0.0518 - 0.159i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-1.28 + 2.22i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.00 + 9.26i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−9.881343466503605950650784517748, −9.621635987437499663564087108011, −8.828991913413052695532460486533, −7.43350416427868161600041416409, −6.59639712908161871813391651183, −6.00869408423735797408221290757, −4.77974653869184324215620691484, −3.97294513194756407832290937547, −2.53476279631280966868421934993, −1.59609042598802423715411656405,
1.89358926134049588639736239173, 2.56228700958877248330851801062, 3.22793409025545097493753417756, 4.93558119729586029314472208512, 5.63910781236649877185614643584, 6.74206197391581343028495138326, 7.963580217109930640398956085770, 8.412690110963803303000535135019, 9.238611873851193823760619090759, 10.00976762314425313144612636038
