| L(s) = 1 | + (−12.0 + 7.74i)2-s + (−3.84 + 26.7i)3-s + (32.0 − 70.1i)4-s + (−296. + 87.0i)5-s + (−160. − 351. i)6-s + (−5.55 − 6.40i)7-s + (−103. − 721. i)8-s + (−699. − 205. i)9-s + (2.89e3 − 3.34e3i)10-s + (323. + 207. i)11-s + (1.75e3 + 1.12e3i)12-s + (−778. + 898. i)13-s + (116. + 34.2i)14-s + (−1.18e3 − 8.26e3i)15-s + (1.32e4 + 1.53e4i)16-s + (−1.39e4 − 3.06e4i)17-s + ⋯ |
| L(s) = 1 | + (−1.06 + 0.684i)2-s + (−0.0821 + 0.571i)3-s + (0.250 − 0.548i)4-s + (−1.06 + 0.311i)5-s + (−0.303 − 0.664i)6-s + (−0.00611 − 0.00706i)7-s + (−0.0716 − 0.498i)8-s + (−0.319 − 0.0939i)9-s + (0.916 − 1.05i)10-s + (0.0732 + 0.0470i)11-s + (0.292 + 0.188i)12-s + (−0.0983 + 0.113i)13-s + (0.0113 + 0.00333i)14-s + (−0.0908 − 0.632i)15-s + (0.811 + 0.936i)16-s + (−0.690 − 1.51i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.482245 + 0.236307i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.482245 + 0.236307i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (3.84 - 26.7i)T \) |
| 23 | \( 1 + (4.58e4 - 3.60e4i)T \) |
| good | 2 | \( 1 + (12.0 - 7.74i)T + (53.1 - 116. i)T^{2} \) |
| 5 | \( 1 + (296. - 87.0i)T + (6.57e4 - 4.22e4i)T^{2} \) |
| 7 | \( 1 + (5.55 + 6.40i)T + (-1.17e5 + 8.15e5i)T^{2} \) |
| 11 | \( 1 + (-323. - 207. i)T + (8.09e6 + 1.77e7i)T^{2} \) |
| 13 | \( 1 + (778. - 898. i)T + (-8.93e6 - 6.21e7i)T^{2} \) |
| 17 | \( 1 + (1.39e4 + 3.06e4i)T + (-2.68e8 + 3.10e8i)T^{2} \) |
| 19 | \( 1 + (-1.99e4 + 4.36e4i)T + (-5.85e8 - 6.75e8i)T^{2} \) |
| 29 | \( 1 + (-2.94e4 - 6.45e4i)T + (-1.12e10 + 1.30e10i)T^{2} \) |
| 31 | \( 1 + (-4.02e4 - 2.80e5i)T + (-2.63e10 + 7.75e9i)T^{2} \) |
| 37 | \( 1 + (-2.53e5 - 7.45e4i)T + (7.98e10 + 5.13e10i)T^{2} \) |
| 41 | \( 1 + (3.76e4 - 1.10e4i)T + (1.63e11 - 1.05e11i)T^{2} \) |
| 43 | \( 1 + (6.12e4 - 4.25e5i)T + (-2.60e11 - 7.65e10i)T^{2} \) |
| 47 | \( 1 - 1.23e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (3.71e5 + 4.28e5i)T + (-1.67e11 + 1.16e12i)T^{2} \) |
| 59 | \( 1 + (8.76e5 - 1.01e6i)T + (-3.54e11 - 2.46e12i)T^{2} \) |
| 61 | \( 1 + (-2.03e5 - 1.41e6i)T + (-3.01e12 + 8.85e11i)T^{2} \) |
| 67 | \( 1 + (-3.45e6 + 2.22e6i)T + (2.51e12 - 5.51e12i)T^{2} \) |
| 71 | \( 1 + (1.54e6 - 9.93e5i)T + (3.77e12 - 8.27e12i)T^{2} \) |
| 73 | \( 1 + (-2.43e6 + 5.33e6i)T + (-7.23e12 - 8.34e12i)T^{2} \) |
| 79 | \( 1 + (-2.68e6 + 3.09e6i)T + (-2.73e12 - 1.90e13i)T^{2} \) |
| 83 | \( 1 + (-5.07e6 - 1.49e6i)T + (2.28e13 + 1.46e13i)T^{2} \) |
| 89 | \( 1 + (-7.60e5 + 5.28e6i)T + (-4.24e13 - 1.24e13i)T^{2} \) |
| 97 | \( 1 + (1.53e6 - 4.50e5i)T + (6.79e13 - 4.36e13i)T^{2} \) |
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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
−13.65445527033370073960576062581, −11.96830499938846379208322024292, −11.05799961245432971856815366012, −9.706458525134104491742891544516, −8.834021438651158518347370735234, −7.60634942641061830164213402740, −6.76127422381449712650386915199, −4.77440316607852350923748295562, −3.25636525040684229173522315559, −0.48417092773012626822567286280,
0.67783624443331363295898393422, 2.08163657797758683392322743320, 3.98619096846090341351173542704, 5.98690234297408579438256589233, 7.87541600035803491406073987157, 8.284052297375993243133987997657, 9.743993657037394149711056617980, 10.91314613584917101559487460586, 11.85864474187135828038721614728, 12.63304894009334864248651732619
