| L(s) = 1 | + (−1.23 + 3.80i)2-s + 30.8i·3-s + (−12.9 − 9.40i)4-s + (−44.9 − 32.6i)5-s + (−117. − 38.1i)6-s + (185. − 60.4i)7-s + (51.7 − 37.6i)8-s − 708.·9-s + (179. − 130. i)10-s + (−365. − 503. i)11-s + (290. − 399. i)12-s + (−207. − 67.3i)13-s + 781. i·14-s + (1.00e3 − 1.38e3i)15-s + (79.1 + 243. i)16-s + (169. + 233. i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + 1.97i·3-s + (−0.404 − 0.293i)4-s + (−0.804 − 0.584i)5-s + (−1.33 − 0.432i)6-s + (1.43 − 0.465i)7-s + (0.286 − 0.207i)8-s − 2.91·9-s + (0.568 − 0.413i)10-s + (−0.911 − 1.25i)11-s + (0.581 − 0.800i)12-s + (−0.340 − 0.110i)13-s + 1.06i·14-s + (1.15 − 1.59i)15-s + (0.0772 + 0.237i)16-s + (0.142 + 0.195i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.683 + 0.730i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.683 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.248719 - 0.107886i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.248719 - 0.107886i\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + (1.02e4 - 3.26e3i)T \) |
| good | 3 | \( 1 - 30.8iT - 243T^{2} \) |
| 5 | \( 1 + (44.9 + 32.6i)T + (965. + 2.97e3i)T^{2} \) |
| 7 | \( 1 + (-185. + 60.4i)T + (1.35e4 - 9.87e3i)T^{2} \) |
| 11 | \( 1 + (365. + 503. i)T + (-4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (207. + 67.3i)T + (3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-169. - 233. i)T + (-4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (255. - 83.0i)T + (2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (859. - 2.64e3i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (1.83e3 - 2.52e3i)T + (-6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-5.60e3 + 4.06e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (9.37e3 + 6.80e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 43 | \( 1 + (3.06e3 - 9.42e3i)T + (-1.18e8 - 8.64e7i)T^{2} \) |
| 47 | \( 1 + (2.18e4 + 7.09e3i)T + (1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-7.68e3 + 1.05e4i)T + (-1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-9.98e3 + 3.07e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-795. - 2.44e3i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-8.76e3 + 1.20e4i)T + (-4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-4.87e3 - 6.71e3i)T + (-5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + 1.29e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.52e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 1.64e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-1.01e4 + 3.31e3i)T + (4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (54.7 - 75.3i)T + (-2.65e9 - 8.16e9i)T^{2} \) |
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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
−13.75002736021023789883570316840, −11.59030209445041358512872381874, −10.90500646009956355953819286819, −9.854164011770445610975747496200, −8.381450190354691968142575051490, −8.128581982787624774222770754116, −5.47901957451884889137904631086, −4.77491617504368145005135852532, −3.64911967270748816849392155378, −0.12102434983979950918544105892,
1.67276916709345886596280857664, 2.63162948328908834940512109241, 5.03794350652150095241790367309, 6.94418468672331027435037803137, 7.81530774407674494123293889370, 8.467714833026816595808853441568, 10.60317473487016227957932891726, 11.89672293282845370340891802213, 11.97742536863947208841122171042, 13.23032332098215065038261496990
