| L(s) = 1 | + (7.51 − 2.73i)2-s + (−13.9 + 79.3i)3-s + (49.0 − 41.1i)4-s + (−397. − 333. i)5-s + (111. + 634. i)6-s + (170. + 294. i)7-s + (256. − 443. i)8-s + (−4.03e3 − 1.46e3i)9-s + (−3.90e3 − 1.42e3i)10-s + (1.83e3 − 3.17e3i)11-s + (2.57e3 + 4.46e3i)12-s + (−1.16e3 − 6.62e3i)13-s + (2.08e3 + 1.75e3i)14-s + (3.20e4 − 2.68e4i)15-s + (711. − 4.03e3i)16-s + (−1.39e4 + 5.08e3i)17-s + ⋯ |
| L(s) = 1 | + (0.664 − 0.241i)2-s + (−0.299 + 1.69i)3-s + (0.383 − 0.321i)4-s + (−1.42 − 1.19i)5-s + (0.211 + 1.19i)6-s + (0.187 + 0.324i)7-s + (0.176 − 0.306i)8-s + (−1.84 − 0.672i)9-s + (−1.23 − 0.449i)10-s + (0.415 − 0.719i)11-s + (0.430 + 0.745i)12-s + (−0.147 − 0.835i)13-s + (0.203 + 0.170i)14-s + (2.44 − 2.05i)15-s + (0.0434 − 0.246i)16-s + (−0.689 + 0.250i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.228413 - 0.439639i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.228413 - 0.439639i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-7.51 + 2.73i)T \) |
| 19 | \( 1 + (1.61e4 + 2.51e4i)T \) |
| good | 3 | \( 1 + (13.9 - 79.3i)T + (-2.05e3 - 747. i)T^{2} \) |
| 5 | \( 1 + (397. + 333. i)T + (1.35e4 + 7.69e4i)T^{2} \) |
| 7 | \( 1 + (-170. - 294. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-1.83e3 + 3.17e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (1.16e3 + 6.62e3i)T + (-5.89e7 + 2.14e7i)T^{2} \) |
| 17 | \( 1 + (1.39e4 - 5.08e3i)T + (3.14e8 - 2.63e8i)T^{2} \) |
| 23 | \( 1 + (5.86e4 - 4.92e4i)T + (5.91e8 - 3.35e9i)T^{2} \) |
| 29 | \( 1 + (1.61e5 + 5.88e4i)T + (1.32e10 + 1.10e10i)T^{2} \) |
| 31 | \( 1 + (8.16e3 + 1.41e4i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + 402.T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-1.57e3 + 8.95e3i)T + (-1.83e11 - 6.66e10i)T^{2} \) |
| 43 | \( 1 + (-6.99e5 - 5.87e5i)T + (4.72e10 + 2.67e11i)T^{2} \) |
| 47 | \( 1 + (1.25e6 + 4.58e5i)T + (3.88e11 + 3.25e11i)T^{2} \) |
| 53 | \( 1 + (1.18e6 - 9.95e5i)T + (2.03e11 - 1.15e12i)T^{2} \) |
| 59 | \( 1 + (-2.26e6 + 8.23e5i)T + (1.90e12 - 1.59e12i)T^{2} \) |
| 61 | \( 1 + (-3.16e5 + 2.65e5i)T + (5.45e11 - 3.09e12i)T^{2} \) |
| 67 | \( 1 + (-5.77e5 - 2.10e5i)T + (4.64e12 + 3.89e12i)T^{2} \) |
| 71 | \( 1 + (-1.42e6 - 1.19e6i)T + (1.57e12 + 8.95e12i)T^{2} \) |
| 73 | \( 1 + (-1.61e5 + 9.17e5i)T + (-1.03e13 - 3.77e12i)T^{2} \) |
| 79 | \( 1 + (-3.86e3 + 2.19e4i)T + (-1.80e13 - 6.56e12i)T^{2} \) |
| 83 | \( 1 + (-6.61e4 - 1.14e5i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (1.16e6 + 6.58e6i)T + (-4.15e13 + 1.51e13i)T^{2} \) |
| 97 | \( 1 + (4.63e5 - 1.68e5i)T + (6.18e13 - 5.19e13i)T^{2} \) |
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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
−14.84774002309428311540690072125, −12.98536949238842091609192452005, −11.63506423976949325682076110800, −11.07091981070386312452451632787, −9.395843933491210854462925262092, −8.248127313915077438185880381528, −5.55365921663306092614361841406, −4.47253266326155600470776869320, −3.59812302795400609007647033042, −0.17417357375209834049815314790,
2.14825421950323544088540697519, 4.04753706143220573297474101807, 6.47666620194529360670251520082, 7.13814012178574198733318655244, 8.040154661362036434720722966000, 10.99111718268662866375375540696, 11.84821654529546484116987554173, 12.62953449502227899379151598999, 14.13079981542436696996117580677, 14.75092758302288283037106063107
