| L(s) = 1 | + (−7.51 + 2.73i)2-s + (−11.6 + 65.9i)3-s + (49.0 − 41.1i)4-s + (−116. − 97.4i)5-s + (−93.0 − 527. i)6-s + (−468. − 811. i)7-s + (−256. + 443. i)8-s + (−2.16e3 − 787. i)9-s + (1.13e3 + 414. i)10-s + (647. − 1.12e3i)11-s + (2.14e3 + 3.71e3i)12-s + (492. + 2.79e3i)13-s + (5.73e3 + 4.81e3i)14-s + (7.77e3 − 6.52e3i)15-s + (711. − 4.03e3i)16-s + (8.11e3 − 2.95e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 + 0.241i)2-s + (−0.248 + 1.41i)3-s + (0.383 − 0.321i)4-s + (−0.415 − 0.348i)5-s + (−0.175 − 0.997i)6-s + (−0.516 − 0.893i)7-s + (−0.176 + 0.306i)8-s + (−0.989 − 0.360i)9-s + (0.360 + 0.131i)10-s + (0.146 − 0.254i)11-s + (0.358 + 0.620i)12-s + (0.0622 + 0.352i)13-s + (0.559 + 0.469i)14-s + (0.595 − 0.499i)15-s + (0.0434 − 0.246i)16-s + (0.400 − 0.145i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 + 0.432i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.901 + 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.727772 - 0.165606i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.727772 - 0.165606i\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + (-2.21e4 + 2.00e4i)T \) |
| good | 3 | \( 1 + (11.6 - 65.9i)T + (-2.05e3 - 747. i)T^{2} \) |
| 5 | \( 1 + (116. + 97.4i)T + (1.35e4 + 7.69e4i)T^{2} \) |
| 7 | \( 1 + (468. + 811. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-647. + 1.12e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-492. - 2.79e3i)T + (-5.89e7 + 2.14e7i)T^{2} \) |
| 17 | \( 1 + (-8.11e3 + 2.95e3i)T + (3.14e8 - 2.63e8i)T^{2} \) |
| 23 | \( 1 + (-1.65e4 + 1.38e4i)T + (5.91e8 - 3.35e9i)T^{2} \) |
| 29 | \( 1 + (-1.41e5 - 5.16e4i)T + (1.32e10 + 1.10e10i)T^{2} \) |
| 31 | \( 1 + (-1.07e4 - 1.86e4i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + 1.12e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-9.09e4 + 5.15e5i)T + (-1.83e11 - 6.66e10i)T^{2} \) |
| 43 | \( 1 + (5.71e5 + 4.79e5i)T + (4.72e10 + 2.67e11i)T^{2} \) |
| 47 | \( 1 + (9.59e5 + 3.49e5i)T + (3.88e11 + 3.25e11i)T^{2} \) |
| 53 | \( 1 + (-1.44e6 + 1.20e6i)T + (2.03e11 - 1.15e12i)T^{2} \) |
| 59 | \( 1 + (-1.22e6 + 4.45e5i)T + (1.90e12 - 1.59e12i)T^{2} \) |
| 61 | \( 1 + (-5.58e5 + 4.68e5i)T + (5.45e11 - 3.09e12i)T^{2} \) |
| 67 | \( 1 + (2.60e6 + 9.47e5i)T + (4.64e12 + 3.89e12i)T^{2} \) |
| 71 | \( 1 + (1.48e6 + 1.24e6i)T + (1.57e12 + 8.95e12i)T^{2} \) |
| 73 | \( 1 + (-7.09e5 + 4.02e6i)T + (-1.03e13 - 3.77e12i)T^{2} \) |
| 79 | \( 1 + (2.32e4 - 1.31e5i)T + (-1.80e13 - 6.56e12i)T^{2} \) |
| 83 | \( 1 + (-4.43e6 - 7.67e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (1.82e5 + 1.03e6i)T + (-4.15e13 + 1.51e13i)T^{2} \) |
| 97 | \( 1 + (1.19e7 - 4.33e6i)T + (6.18e13 - 5.19e13i)T^{2} \) |
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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
−15.14373944183638638086319994784, −13.77161507753571691563860694391, −11.85749051891228440173652657486, −10.61444447175626854991488447063, −9.802057009410058511400216368581, −8.609289702137225132527708597523, −6.90370307486908355299503716730, −5.01774826398593710570616740901, −3.61038450674901328517152564877, −0.47499159987833571770477564626,
1.28225930548655543971369548478, 2.93160786582753887851612658711, 5.98078107380808325886187952366, 7.20039359266971122362156126893, 8.261940130651208035715773191376, 9.836129836230726490657164513507, 11.54787757562404938139460311822, 12.25093594313980181281061770239, 13.28208519276046348939456839338, 14.94216999111827724016176876580
