| L(s) = 1 | + (−9.79 − 5.65i)2-s + (−127. − 73.6i)3-s + (63.9 + 110. i)4-s + (381. − 660. i)5-s + (833. + 1.44e3i)6-s + 2.76e3·7-s − 1.44e3i·8-s + (7.58e3 + 1.31e4i)9-s + (−7.46e3 + 4.31e3i)10-s + 1.18e4·11-s − 1.88e4i·12-s + (4.22e4 − 2.43e4i)13-s + (−2.70e4 − 1.56e4i)14-s + (−9.73e4 + 5.61e4i)15-s + (−8.19e3 + 1.41e4i)16-s + (2.00e4 − 3.47e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (−1.57 − 0.909i)3-s + (0.249 + 0.433i)4-s + (0.609 − 1.05i)5-s + (0.643 + 1.11i)6-s + 1.15·7-s − 0.353i·8-s + (1.15 + 2.00i)9-s + (−0.746 + 0.431i)10-s + 0.812·11-s − 0.909i·12-s + (1.47 − 0.853i)13-s + (−0.704 − 0.406i)14-s + (−1.92 + 1.10i)15-s + (−0.125 + 0.216i)16-s + (0.240 − 0.416i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.511 + 0.859i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.511 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.572379 - 1.00662i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.572379 - 1.00662i\) |
| \(L(5)\) |
|
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 | \( 1 + (5.22e4 + 1.19e5i)T \) |
| good | 3 | \( 1 + (127. + 73.6i)T + (3.28e3 + 5.68e3i)T^{2} \) |
| 5 | \( 1 + (-381. + 660. i)T + (-1.95e5 - 3.38e5i)T^{2} \) |
| 7 | \( 1 - 2.76e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 1.18e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + (-4.22e4 + 2.43e4i)T + (4.07e8 - 7.06e8i)T^{2} \) |
| 17 | \( 1 + (-2.00e4 + 3.47e4i)T + (-3.48e9 - 6.04e9i)T^{2} \) |
| 23 | \( 1 + (-2.61e5 - 4.52e5i)T + (-3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + (-3.68e5 + 2.13e5i)T + (2.50e11 - 4.33e11i)T^{2} \) |
| 31 | \( 1 - 2.24e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 + 1.20e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + (-2.87e6 - 1.66e6i)T + (3.99e12 + 6.91e12i)T^{2} \) |
| 43 | \( 1 + (9.34e4 - 1.61e5i)T + (-5.84e12 - 1.01e13i)T^{2} \) |
| 47 | \( 1 + (-2.98e6 - 5.17e6i)T + (-1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-2.46e6 + 1.42e6i)T + (3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (1.06e7 + 6.13e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (9.48e6 + 1.64e7i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (2.63e7 - 1.52e7i)T + (2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + (7.06e6 + 4.08e6i)T + (3.22e14 + 5.59e14i)T^{2} \) |
| 73 | \( 1 + (-1.13e7 + 1.95e7i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-8.18e6 - 4.72e6i)T + (7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + 3.04e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + (-6.05e6 + 3.49e6i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 + (-8.69e7 - 5.02e7i)T + (3.91e15 + 6.78e15i)T^{2} \) |
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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
−13.55636901925450392694876029795, −12.69553598376592273780506842706, −11.51065753376641874204352161905, −10.90032979693788807375294343151, −9.048891954376255319460963026140, −7.65912179284202313003037186736, −6.06588861819375984432033133808, −4.92381483732612637997310940127, −1.42040949789237196274148725741, −0.941802251666644627116947969675,
1.29877215463356525214375286711, 4.29257304433525016189222046265, 5.93357877076666141670253508358, 6.65967497179685859274622281453, 8.850350187276451879639065618278, 10.43941406150007800002874241964, 10.86295408458445009177238386005, 11.89861636084493978882197764252, 14.25832960700805220800230245779, 15.07373503303966249850185490601
