| L(s) = 1 | + (−1.23 + 3.80i)2-s + 10.4i·3-s + (−12.9 − 9.40i)4-s + (16.4 + 11.9i)5-s + (−39.8 − 12.9i)6-s + (166. − 54.1i)7-s + (51.7 − 37.6i)8-s + 133.·9-s + (−65.7 + 47.7i)10-s + (442. + 609. i)11-s + (98.4 − 135. i)12-s + (−638. − 207. i)13-s + 700. i·14-s + (−125. + 172. i)15-s + (79.1 + 243. i)16-s + (−166. − 229. i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + 0.671i·3-s + (−0.404 − 0.293i)4-s + (0.294 + 0.213i)5-s + (−0.451 − 0.146i)6-s + (1.28 − 0.417i)7-s + (0.286 − 0.207i)8-s + 0.549·9-s + (−0.208 + 0.151i)10-s + (1.10 + 1.51i)11-s + (0.197 − 0.271i)12-s + (−1.04 − 0.340i)13-s + 0.955i·14-s + (−0.143 + 0.197i)15-s + (0.0772 + 0.237i)16-s + (−0.140 − 0.192i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.09363 + 1.56246i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09363 + 1.56246i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 - 3.80i)T \) |
| 41 | \( 1 + (-1.07e4 - 1.02e3i)T \) |
| good | 3 | \( 1 - 10.4iT - 243T^{2} \) |
| 5 | \( 1 + (-16.4 - 11.9i)T + (965. + 2.97e3i)T^{2} \) |
| 7 | \( 1 + (-166. + 54.1i)T + (1.35e4 - 9.87e3i)T^{2} \) |
| 11 | \( 1 + (-442. - 609. i)T + (-4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (638. + 207. i)T + (3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (166. + 229. i)T + (-4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-719. + 233. i)T + (2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (688. - 2.11e3i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (1.84e3 - 2.54e3i)T + (-6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (3.30e3 - 2.40e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-9.30e3 - 6.76e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 43 | \( 1 + (1.08e3 - 3.34e3i)T + (-1.18e8 - 8.64e7i)T^{2} \) |
| 47 | \( 1 + (6.07e3 + 1.97e3i)T + (1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (182. - 251. i)T + (-1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-1.09e4 + 3.37e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (2.56e3 + 7.89e3i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-4.23e4 + 5.82e4i)T + (-4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-4.17e4 - 5.74e4i)T + (-5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + 7.25e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.55e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 3.00e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (1.32e5 - 4.30e4i)T + (4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (5.55e3 - 7.65e3i)T + (-2.65e9 - 8.16e9i)T^{2} \) |
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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
−14.24615506770026818078496486620, −12.65446738630814964036039130051, −11.34237442425486816076270265196, −10.00505221431369271952302686747, −9.453557580949052622762572968137, −7.76671237472382125806583207127, −6.93401475586873496954947499190, −5.05942488348913571745935513959, −4.26343013951847869534612221540, −1.62388483571441802228059074065,
1.00461893350628321502640101710, 2.15558373469654889293167144599, 4.20907568229171416609162540789, 5.78404146658155671325263934879, 7.44454235710441215016104202929, 8.560691659655359201054406626179, 9.599158654315053702207173311217, 11.16465858204038168804031220167, 11.80710869530419186325422000671, 12.87434149240263078675971777878
