| L(s) = 1 | + (9.79 + 5.65i)2-s + (−3.23 − 1.86i)3-s + (63.9 + 110. i)4-s + (−499. + 864. i)5-s + (−21.1 − 36.5i)6-s + 3.10e3·7-s + 1.44e3i·8-s + (−3.27e3 − 5.66e3i)9-s + (−9.78e3 + 5.64e3i)10-s − 2.37e4·11-s − 478. i·12-s + (−4.57e3 + 2.63e3i)13-s + (3.04e4 + 1.75e4i)14-s + (3.22e3 − 1.86e3i)15-s + (−8.19e3 + 1.41e4i)16-s + (−4.60e4 + 7.97e4i)17-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.0399 − 0.0230i)3-s + (0.249 + 0.433i)4-s + (−0.798 + 1.38i)5-s + (−0.0163 − 0.0282i)6-s + 1.29·7-s + 0.353i·8-s + (−0.498 − 0.864i)9-s + (−0.978 + 0.564i)10-s − 1.62·11-s − 0.0230i·12-s + (−0.160 + 0.0924i)13-s + (0.792 + 0.457i)14-s + (0.0637 − 0.0368i)15-s + (−0.125 + 0.216i)16-s + (−0.551 + 0.954i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.234i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.163425 + 1.37355i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.163425 + 1.37355i\) |
| \(L(5)\) |
|
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 | \( 1 + (9.86e4 - 8.51e4i)T \) |
| good | 3 | \( 1 + (3.23 + 1.86i)T + (3.28e3 + 5.68e3i)T^{2} \) |
| 5 | \( 1 + (499. - 864. i)T + (-1.95e5 - 3.38e5i)T^{2} \) |
| 7 | \( 1 - 3.10e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + 2.37e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + (4.57e3 - 2.63e3i)T + (4.07e8 - 7.06e8i)T^{2} \) |
| 17 | \( 1 + (4.60e4 - 7.97e4i)T + (-3.48e9 - 6.04e9i)T^{2} \) |
| 23 | \( 1 + (-9.67e4 - 1.67e5i)T + (-3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + (-1.09e5 + 6.31e4i)T + (2.50e11 - 4.33e11i)T^{2} \) |
| 31 | \( 1 - 1.20e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 + 3.31e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + (-3.62e6 - 2.09e6i)T + (3.99e12 + 6.91e12i)T^{2} \) |
| 43 | \( 1 + (1.80e5 - 3.12e5i)T + (-5.84e12 - 1.01e13i)T^{2} \) |
| 47 | \( 1 + (-3.03e6 - 5.25e6i)T + (-1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (4.95e5 - 2.86e5i)T + (3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-8.86e5 - 5.11e5i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-1.16e7 - 2.01e7i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (1.53e7 - 8.87e6i)T + (2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + (-3.44e7 - 1.98e7i)T + (3.22e14 + 5.59e14i)T^{2} \) |
| 73 | \( 1 + (9.67e6 - 1.67e7i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-1.94e7 - 1.12e7i)T + (7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + 6.28e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + (-3.79e6 + 2.19e6i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 + (-1.48e8 - 8.58e7i)T + (3.91e15 + 6.78e15i)T^{2} \) |
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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
−14.89454925406075557464843832746, −14.36536803524815791500278545743, −12.70402351342480189378890719588, −11.34973197170892949019896841190, −10.69757985319182206210356349477, −8.263014377742839726293166977875, −7.33494084136219195333170690314, −5.82394583776731301847245601777, −4.08503520520725518768389940848, −2.57801189079960270995433711317,
0.42832487874291901827809471851, 2.33149573647393660346615229249, 4.76607895540606726190194708614, 5.03975323001766780306343553207, 7.75360935483060519746526768234, 8.627438044604043639039450524951, 10.72672586303634590806044979561, 11.59686417757764609143351636408, 12.81009462520581896350384528669, 13.73133564333595293397231043835
