| L(s) = 1 | + (1.38 − 7.87i)2-s + (−46.9 − 39.4i)3-s + (−60.1 − 21.8i)4-s + (−248. + 90.5i)5-s + (−375. + 315. i)6-s + (81.2 + 140. i)7-s + (−256 + 443. i)8-s + (273. + 1.54e3i)9-s + (367. + 2.08e3i)10-s + (926. − 1.60e3i)11-s + (1.96e3 + 3.39e3i)12-s + (3.43e3 − 2.88e3i)13-s + (1.22e3 − 444. i)14-s + (1.52e4 + 5.55e3i)15-s + (3.13e3 + 2.63e3i)16-s + (2.63 − 14.9i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (−1.00 − 0.842i)3-s + (−0.469 − 0.171i)4-s + (−0.890 + 0.324i)5-s + (−0.710 + 0.596i)6-s + (0.0895 + 0.155i)7-s + (−0.176 + 0.306i)8-s + (0.124 + 0.708i)9-s + (0.116 + 0.659i)10-s + (0.209 − 0.363i)11-s + (0.327 + 0.567i)12-s + (0.433 − 0.363i)13-s + (0.119 − 0.0433i)14-s + (1.16 + 0.424i)15-s + (0.191 + 0.160i)16-s + (0.000129 − 0.000736i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.438028 + 0.134716i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.438028 + 0.134716i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.38 + 7.87i)T \) |
| 19 | \( 1 + (-2.65e3 - 2.97e4i)T \) |
| good | 3 | \( 1 + (46.9 + 39.4i)T + (379. + 2.15e3i)T^{2} \) |
| 5 | \( 1 + (248. - 90.5i)T + (5.98e4 - 5.02e4i)T^{2} \) |
| 7 | \( 1 + (-81.2 - 140. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-926. + 1.60e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-3.43e3 + 2.88e3i)T + (1.08e7 - 6.17e7i)T^{2} \) |
| 17 | \( 1 + (-2.63 + 14.9i)T + (-3.85e8 - 1.40e8i)T^{2} \) |
| 23 | \( 1 + (-1.80e4 - 6.57e3i)T + (2.60e9 + 2.18e9i)T^{2} \) |
| 29 | \( 1 + (-2.27e4 - 1.28e5i)T + (-1.62e10 + 5.89e9i)T^{2} \) |
| 31 | \( 1 + (-7.48e4 - 1.29e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + 1.67e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (3.21e5 + 2.69e5i)T + (3.38e10 + 1.91e11i)T^{2} \) |
| 43 | \( 1 + (7.32e5 - 2.66e5i)T + (2.08e11 - 1.74e11i)T^{2} \) |
| 47 | \( 1 + (8.85e3 + 5.02e4i)T + (-4.76e11 + 1.73e11i)T^{2} \) |
| 53 | \( 1 + (2.89e5 + 1.05e5i)T + (8.99e11 + 7.55e11i)T^{2} \) |
| 59 | \( 1 + (3.28e5 - 1.86e6i)T + (-2.33e12 - 8.51e11i)T^{2} \) |
| 61 | \( 1 + (-2.83e6 - 1.03e6i)T + (2.40e12 + 2.02e12i)T^{2} \) |
| 67 | \( 1 + (-5.54e5 - 3.14e6i)T + (-5.69e12 + 2.07e12i)T^{2} \) |
| 71 | \( 1 + (1.33e6 - 4.85e5i)T + (6.96e12 - 5.84e12i)T^{2} \) |
| 73 | \( 1 + (-1.59e6 - 1.34e6i)T + (1.91e12 + 1.08e13i)T^{2} \) |
| 79 | \( 1 + (5.98e5 + 5.02e5i)T + (3.33e12 + 1.89e13i)T^{2} \) |
| 83 | \( 1 + (2.74e6 + 4.75e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-4.58e6 + 3.84e6i)T + (7.68e12 - 4.35e13i)T^{2} \) |
| 97 | \( 1 + (-3.05e5 + 1.73e6i)T + (-7.59e13 - 2.76e13i)T^{2} \) |
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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
−14.76225393670734450877015034506, −13.30596730373742007459272362471, −12.14381094470496198930525934007, −11.54961755960459053455378750580, −10.45367920446286986923977789715, −8.439016439378622658307363307058, −6.93780678417854405200302183783, −5.47351018879401651392160204562, −3.51536450195069393976030988967, −1.25262614718390567624838262506,
0.25438306860602687805955330004, 4.06177306975996321899306602440, 5.01340892514907871175813287168, 6.57949832603821799579575013406, 8.145828018128214805335161296009, 9.660796384269269610736919080389, 11.12840704317324990937590587164, 12.03259383231868438711482482203, 13.58499245526580445456303383348, 15.21356426481023045565506722581
