| L(s) = 1 | + (9.79 + 5.65i)2-s + (−107. − 61.8i)3-s + (63.9 + 110. i)4-s + (333. − 577. i)5-s + (−699. − 1.21e3i)6-s − 377.·7-s + 1.44e3i·8-s + (4.35e3 + 7.55e3i)9-s + (6.52e3 − 3.76e3i)10-s − 7.60e3·11-s − 1.58e4i·12-s + (−1.54e4 + 8.94e3i)13-s + (−3.70e3 − 2.13e3i)14-s + (−7.13e4 + 4.11e4i)15-s + (−8.19e3 + 1.41e4i)16-s + (−6.17e4 + 1.06e5i)17-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (−1.32 − 0.763i)3-s + (0.249 + 0.433i)4-s + (0.533 − 0.923i)5-s + (−0.539 − 0.934i)6-s − 0.157·7-s + 0.353i·8-s + (0.664 + 1.15i)9-s + (0.652 − 0.376i)10-s − 0.519·11-s − 0.763i·12-s + (−0.542 + 0.313i)13-s + (−0.0963 − 0.0556i)14-s + (−1.40 + 0.813i)15-s + (−0.125 + 0.216i)16-s + (−0.739 + 1.28i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.848 - 0.529i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0458559 + 0.160133i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0458559 + 0.160133i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-9.79 - 5.65i)T \) |
| 19 | \( 1 + (1.20e5 - 4.97e4i)T \) |
| good | 3 | \( 1 + (107. + 61.8i)T + (3.28e3 + 5.68e3i)T^{2} \) |
| 5 | \( 1 + (-333. + 577. i)T + (-1.95e5 - 3.38e5i)T^{2} \) |
| 7 | \( 1 + 377.T + 5.76e6T^{2} \) |
| 11 | \( 1 + 7.60e3T + 2.14e8T^{2} \) |
| 13 | \( 1 + (1.54e4 - 8.94e3i)T + (4.07e8 - 7.06e8i)T^{2} \) |
| 17 | \( 1 + (6.17e4 - 1.06e5i)T + (-3.48e9 - 6.04e9i)T^{2} \) |
| 23 | \( 1 + (7.23e4 + 1.25e5i)T + (-3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + (-1.31e5 + 7.61e4i)T + (2.50e11 - 4.33e11i)T^{2} \) |
| 31 | \( 1 + 2.11e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 3.04e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + (1.11e6 + 6.43e5i)T + (3.99e12 + 6.91e12i)T^{2} \) |
| 43 | \( 1 + (-7.54e5 + 1.30e6i)T + (-5.84e12 - 1.01e13i)T^{2} \) |
| 47 | \( 1 + (2.82e6 + 4.89e6i)T + (-1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-9.50e6 + 5.48e6i)T + (3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (8.35e6 + 4.82e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-7.02e6 - 1.21e7i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (2.98e7 - 1.72e7i)T + (2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + (5.30e6 + 3.06e6i)T + (3.22e14 + 5.59e14i)T^{2} \) |
| 73 | \( 1 + (-2.72e7 + 4.72e7i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (5.50e7 + 3.18e7i)T + (7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + 1.25e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + (8.39e7 - 4.84e7i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 + (6.41e7 + 3.70e7i)T + (3.91e15 + 6.78e15i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.06094012813990021892903277934, −13.32806894513619215327249395167, −12.76651096621772509457683032895, −11.81612566377949071125724675262, −10.39541018479638740021849850692, −8.420660559342611291770771959508, −6.74068378698435963459521040844, −5.77681382579327054028397258835, −4.62930725518169514261815906755, −1.77250510297040357712669430986,
0.05974590467694530485495945611, 2.65594552923080741395396755097, 4.56907929220170453376765884484, 5.73303817783257531968053260273, 6.89487008863854730756240726202, 9.674236253429827302838165355993, 10.64107705947818798777341181535, 11.34678697489409218711797258946, 12.67920627161221008082284125273, 14.05322778232352179208608532598
