| 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
−14.94216999111827724016176876580, −13.28208519276046348939456839338, −12.25093594313980181281061770239, −11.54787757562404938139460311822, −9.836129836230726490657164513507, −8.261940130651208035715773191376, −7.20039359266971122362156126893, −5.98078107380808325886187952366, −2.93160786582753887851612658711, −1.28225930548655543971369548478,
0.47499159987833571770477564626, 3.61038450674901328517152564877, 5.01774826398593710570616740901, 6.90370307486908355299503716730, 8.609289702137225132527708597523, 9.802057009410058511400216368581, 10.61444447175626854991488447063, 11.85749051891228440173652657486, 13.77161507753571691563860694391, 15.14373944183638638086319994784
