| L(s) = 1 | + (1.38 + 7.87i)2-s + (−21.2 + 17.7i)3-s + (−60.1 + 21.8i)4-s + (194. + 70.8i)5-s + (−169. − 142. i)6-s + (−485. + 841. i)7-s + (−256 − 443. i)8-s + (−246. + 1.39e3i)9-s + (−287. + 1.63e3i)10-s + (−2.51e3 − 4.35e3i)11-s + (885. − 1.53e3i)12-s + (−5.80e3 − 4.87e3i)13-s + (−7.30e3 − 2.65e3i)14-s + (−5.39e3 + 1.96e3i)15-s + (3.13e3 − 2.63e3i)16-s + (−3.73e3 − 2.11e4i)17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (−0.453 + 0.380i)3-s + (−0.469 + 0.171i)4-s + (0.696 + 0.253i)5-s + (−0.320 − 0.269i)6-s + (−0.535 + 0.926i)7-s + (−0.176 − 0.306i)8-s + (−0.112 + 0.639i)9-s + (−0.0910 + 0.516i)10-s + (−0.569 − 0.987i)11-s + (0.147 − 0.256i)12-s + (−0.732 − 0.614i)13-s + (−0.711 − 0.258i)14-s + (−0.412 + 0.150i)15-s + (0.191 − 0.160i)16-s + (−0.184 − 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 + 0.685i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.728 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.208149 - 0.524746i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.208149 - 0.524746i\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + (-2.84e4 - 9.05e3i)T \) |
| good | 3 | \( 1 + (21.2 - 17.7i)T + (379. - 2.15e3i)T^{2} \) |
| 5 | \( 1 + (-194. - 70.8i)T + (5.98e4 + 5.02e4i)T^{2} \) |
| 7 | \( 1 + (485. - 841. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (2.51e3 + 4.35e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (5.80e3 + 4.87e3i)T + (1.08e7 + 6.17e7i)T^{2} \) |
| 17 | \( 1 + (3.73e3 + 2.11e4i)T + (-3.85e8 + 1.40e8i)T^{2} \) |
| 23 | \( 1 + (7.38e4 - 2.68e4i)T + (2.60e9 - 2.18e9i)T^{2} \) |
| 29 | \( 1 + (1.16e4 - 6.58e4i)T + (-1.62e10 - 5.89e9i)T^{2} \) |
| 31 | \( 1 + (1.47e5 - 2.56e5i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 - 1.56e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (1.13e5 - 9.53e4i)T + (3.38e10 - 1.91e11i)T^{2} \) |
| 43 | \( 1 + (-4.01e5 - 1.46e5i)T + (2.08e11 + 1.74e11i)T^{2} \) |
| 47 | \( 1 + (1.70e5 - 9.64e5i)T + (-4.76e11 - 1.73e11i)T^{2} \) |
| 53 | \( 1 + (-1.34e6 + 4.88e5i)T + (8.99e11 - 7.55e11i)T^{2} \) |
| 59 | \( 1 + (-6.16e3 - 3.49e4i)T + (-2.33e12 + 8.51e11i)T^{2} \) |
| 61 | \( 1 + (2.16e6 - 7.89e5i)T + (2.40e12 - 2.02e12i)T^{2} \) |
| 67 | \( 1 + (1.49e5 - 8.45e5i)T + (-5.69e12 - 2.07e12i)T^{2} \) |
| 71 | \( 1 + (4.46e6 + 1.62e6i)T + (6.96e12 + 5.84e12i)T^{2} \) |
| 73 | \( 1 + (-9.97e5 + 8.37e5i)T + (1.91e12 - 1.08e13i)T^{2} \) |
| 79 | \( 1 + (2.74e6 - 2.30e6i)T + (3.33e12 - 1.89e13i)T^{2} \) |
| 83 | \( 1 + (-1.38e6 + 2.39e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-4.83e6 - 4.05e6i)T + (7.68e12 + 4.35e13i)T^{2} \) |
| 97 | \( 1 + (-4.39e5 - 2.49e6i)T + (-7.59e13 + 2.76e13i)T^{2} \) |
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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
−15.94613024957319750873452689015, −14.32562861338200595115276204806, −13.40146143079425417257475068373, −11.97708480995876891282103028302, −10.42846700806491350248528160196, −9.301029452622397371899266097344, −7.73264724645315980710991646167, −5.92331938031719933829414724020, −5.24602336143429842971489456723, −2.81476791092542288153951146703,
0.23816988163387082816754669204, 1.95622148973613467740843947321, 4.08517130744229775182362610900, 5.84080581346946599746601843546, 7.32114859618492158522965406368, 9.453951686216509163776667045266, 10.22128810869961718539933137415, 11.77515523673062737341790572547, 12.78993096953154200309323711238, 13.64642796907026964769668369777
