| L(s) = 1 | + (−1.10 − 1.38i)2-s + (−5.76 − 0.869i)3-s + (6.42 − 28.1i)4-s + (43.6 + 29.7i)5-s + (5.15 + 8.93i)6-s + (20.6 − 35.7i)7-s + (−96.9 + 46.7i)8-s + (−199. − 61.5i)9-s + (−6.98 − 93.2i)10-s + (−168. − 739. i)11-s + (−61.5 + 156. i)12-s + (11.5 − 153. i)13-s + (−72.2 + 10.8i)14-s + (−226. − 209. i)15-s + (−660. − 318. i)16-s + (920. − 627. i)17-s + ⋯ |
| L(s) = 1 | + (−0.194 − 0.244i)2-s + (−0.370 − 0.0557i)3-s + (0.200 − 0.879i)4-s + (0.781 + 0.532i)5-s + (0.0585 + 0.101i)6-s + (0.159 − 0.275i)7-s + (−0.535 + 0.258i)8-s + (−0.821 − 0.253i)9-s + (−0.0220 − 0.294i)10-s + (−0.420 − 1.84i)11-s + (−0.123 + 0.314i)12-s + (0.0189 − 0.252i)13-s + (−0.0984 + 0.0148i)14-s + (−0.259 − 0.240i)15-s + (−0.645 − 0.310i)16-s + (0.772 − 0.526i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.540 + 0.841i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.540 + 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.555429 - 1.01698i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.555429 - 1.01698i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-1.06e4 - 5.83e3i)T \) |
| good | 2 | \( 1 + (1.10 + 1.38i)T + (-7.12 + 31.1i)T^{2} \) |
| 3 | \( 1 + (5.76 + 0.869i)T + (232. + 71.6i)T^{2} \) |
| 5 | \( 1 + (-43.6 - 29.7i)T + (1.14e3 + 2.90e3i)T^{2} \) |
| 7 | \( 1 + (-20.6 + 35.7i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (168. + 739. i)T + (-1.45e5 + 6.98e4i)T^{2} \) |
| 13 | \( 1 + (-11.5 + 153. i)T + (-3.67e5 - 5.53e4i)T^{2} \) |
| 17 | \( 1 + (-920. + 627. i)T + (5.18e5 - 1.32e6i)T^{2} \) |
| 19 | \( 1 + (1.08e3 - 333. i)T + (2.04e6 - 1.39e6i)T^{2} \) |
| 23 | \( 1 + (2.05e3 - 1.90e3i)T + (4.80e5 - 6.41e6i)T^{2} \) |
| 29 | \( 1 + (-8.02e3 + 1.20e3i)T + (1.95e7 - 6.04e6i)T^{2} \) |
| 31 | \( 1 + (-954. + 2.43e3i)T + (-2.09e7 - 1.94e7i)T^{2} \) |
| 37 | \( 1 + (-3.34e3 - 5.79e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + (2.43e3 + 3.05e3i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 47 | \( 1 + (-5.75e3 + 2.52e4i)T + (-2.06e8 - 9.95e7i)T^{2} \) |
| 53 | \( 1 + (360. + 4.81e3i)T + (-4.13e8 + 6.23e7i)T^{2} \) |
| 59 | \( 1 + (4.02e4 + 1.93e4i)T + (4.45e8 + 5.58e8i)T^{2} \) |
| 61 | \( 1 + (-9.82e3 - 2.50e4i)T + (-6.19e8 + 5.74e8i)T^{2} \) |
| 67 | \( 1 + (-2.43e3 + 749. i)T + (1.11e9 - 7.60e8i)T^{2} \) |
| 71 | \( 1 + (-4.05e4 - 3.75e4i)T + (1.34e8 + 1.79e9i)T^{2} \) |
| 73 | \( 1 + (3.76e3 - 5.02e4i)T + (-2.04e9 - 3.08e8i)T^{2} \) |
| 79 | \( 1 + (-4.31e3 + 7.47e3i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-1.29e4 - 1.95e3i)T + (3.76e9 + 1.16e9i)T^{2} \) |
| 89 | \( 1 + (4.78e4 + 7.20e3i)T + (5.33e9 + 1.64e9i)T^{2} \) |
| 97 | \( 1 + (1.05e4 + 4.60e4i)T + (-7.73e9 + 3.72e9i)T^{2} \) |
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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
−14.21622584795551856061304698084, −13.82372904090014895595144175140, −11.77683625819296111955293674169, −10.83299611611946171083231492106, −9.965348052190606961697358499987, −8.409470789118498098569742216761, −6.25790912995454237353404110497, −5.61187044022889740638888513778, −2.79353836464551545377456438315, −0.68430125672089788524182825993,
2.30609503678903951823336360052, 4.71184184089488045515163174367, 6.26740282571047358433491198150, 7.84477350976558160345492987927, 9.051598167629061607324575604224, 10.43355924864909719380748869672, 12.12808272977352921703865904817, 12.66699495204786856854795357534, 14.20407800388739721229387004504, 15.55354470368070301747995235409
