| L(s) = 1 | + (−6.47 − 4.70i)2-s + (−28.7 + 20.9i)3-s + (19.7 + 60.8i)4-s − 353.·5-s + 284.·6-s + (378. + 1.16e3i)7-s + (158. − 486. i)8-s + (−284. + 876. i)9-s + (2.28e3 + 1.66e3i)10-s + (418. + 1.28e3i)11-s + (−1.84e3 − 1.33e3i)12-s + (−6.21e3 + 4.51e3i)13-s + (3.02e3 − 9.30e3i)14-s + (1.01e4 − 7.38e3i)15-s + (−3.31e3 + 2.40e3i)16-s + (−1.81e3 + 5.59e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.415i)2-s + (−0.615 + 0.447i)3-s + (0.154 + 0.475i)4-s − 1.26·5-s + 0.537·6-s + (0.416 + 1.28i)7-s + (0.109 − 0.336i)8-s + (−0.130 + 0.400i)9-s + (0.723 + 0.525i)10-s + (0.0948 + 0.291i)11-s + (−0.307 − 0.223i)12-s + (−0.784 + 0.570i)13-s + (0.294 − 0.906i)14-s + (0.778 − 0.565i)15-s + (−0.202 + 0.146i)16-s + (−0.0896 + 0.275i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.508 + 0.861i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.508 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0260092 - 0.0455358i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0260092 - 0.0455358i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (6.47 + 4.70i)T \) |
| 31 | \( 1 + (1.62e5 + 3.35e4i)T \) |
| good | 3 | \( 1 + (28.7 - 20.9i)T + (675. - 2.07e3i)T^{2} \) |
| 5 | \( 1 + 353.T + 7.81e4T^{2} \) |
| 7 | \( 1 + (-378. - 1.16e3i)T + (-6.66e5 + 4.84e5i)T^{2} \) |
| 11 | \( 1 + (-418. - 1.28e3i)T + (-1.57e7 + 1.14e7i)T^{2} \) |
| 13 | \( 1 + (6.21e3 - 4.51e3i)T + (1.93e7 - 5.96e7i)T^{2} \) |
| 17 | \( 1 + (1.81e3 - 5.59e3i)T + (-3.31e8 - 2.41e8i)T^{2} \) |
| 19 | \( 1 + (-2.06e4 - 1.50e4i)T + (2.76e8 + 8.50e8i)T^{2} \) |
| 23 | \( 1 + (-1.15e4 + 3.54e4i)T + (-2.75e9 - 2.00e9i)T^{2} \) |
| 29 | \( 1 + (1.72e5 + 1.25e5i)T + (5.33e9 + 1.64e10i)T^{2} \) |
| 37 | \( 1 + 3.78e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-5.68e5 - 4.13e5i)T + (6.01e10 + 1.85e11i)T^{2} \) |
| 43 | \( 1 + (-2.02e5 - 1.47e5i)T + (8.39e10 + 2.58e11i)T^{2} \) |
| 47 | \( 1 + (-6.72e5 + 4.88e5i)T + (1.56e11 - 4.81e11i)T^{2} \) |
| 53 | \( 1 + (-1.42e5 + 4.39e5i)T + (-9.50e11 - 6.90e11i)T^{2} \) |
| 59 | \( 1 + (-1.13e6 + 8.27e5i)T + (7.69e11 - 2.36e12i)T^{2} \) |
| 61 | \( 1 - 5.19e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.34e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (1.43e6 - 4.40e6i)T + (-7.35e12 - 5.34e12i)T^{2} \) |
| 73 | \( 1 + (9.06e4 + 2.79e5i)T + (-8.93e12 + 6.49e12i)T^{2} \) |
| 79 | \( 1 + (9.60e5 - 2.95e6i)T + (-1.55e13 - 1.12e13i)T^{2} \) |
| 83 | \( 1 + (5.55e6 + 4.03e6i)T + (8.38e12 + 2.58e13i)T^{2} \) |
| 89 | \( 1 + (-2.15e5 - 6.63e5i)T + (-3.57e13 + 2.59e13i)T^{2} \) |
| 97 | \( 1 + (-4.56e5 - 1.40e6i)T + (-6.53e13 + 4.74e13i)T^{2} \) |
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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
−14.59088531514080251439077232969, −12.56731397694635952947389216352, −11.70171836553350161697525970031, −11.18257912031539578593920307083, −9.724799958657718961103946911450, −8.477907296286786170480749735593, −7.42775880229825372083989716240, −5.46648087429180117184310341510, −4.10976976101895579056883692252, −2.23207120019619612253494134736,
0.03310891890548539044988456319, 0.941008776935359914740091516176, 3.70925890225928497336856526126, 5.36166279490317303857315793280, 7.28815866459831795626279799725, 7.41395720558323723752258433908, 9.097628747253860869383108881491, 10.73319466764092408887124955815, 11.45017865073636305262944202730, 12.57288109115734144201579835581
