| L(s) = 1 | + (9.79 + 5.65i)2-s + (61.4 + 35.4i)3-s + (63.9 + 110. i)4-s + (219. − 380. i)5-s + (401. + 695. i)6-s + 2.50e3·7-s + 1.44e3i·8-s + (−761. − 1.31e3i)9-s + (4.31e3 − 2.48e3i)10-s + 2.34e4·11-s + 9.08e3i·12-s + (−1.51e4 + 8.72e3i)13-s + (2.45e4 + 1.41e4i)14-s + (2.70e4 − 1.56e4i)15-s + (−8.19e3 + 1.41e4i)16-s + (−6.06e4 + 1.05e5i)17-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (0.758 + 0.438i)3-s + (0.249 + 0.433i)4-s + (0.351 − 0.609i)5-s + (0.309 + 0.536i)6-s + 1.04·7-s + 0.353i·8-s + (−0.116 − 0.201i)9-s + (0.431 − 0.248i)10-s + 1.59·11-s + 0.438i·12-s + (−0.529 + 0.305i)13-s + (0.638 + 0.368i)14-s + (0.534 − 0.308i)15-s + (−0.125 + 0.216i)16-s + (−0.726 + 1.25i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(3.65141 + 1.33630i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.65141 + 1.33630i\) |
| \(L(5)\) |
|
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 | \( 1 + (1.04e4 + 1.29e5i)T \) |
| good | 3 | \( 1 + (-61.4 - 35.4i)T + (3.28e3 + 5.68e3i)T^{2} \) |
| 5 | \( 1 + (-219. + 380. i)T + (-1.95e5 - 3.38e5i)T^{2} \) |
| 7 | \( 1 - 2.50e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 2.34e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + (1.51e4 - 8.72e3i)T + (4.07e8 - 7.06e8i)T^{2} \) |
| 17 | \( 1 + (6.06e4 - 1.05e5i)T + (-3.48e9 - 6.04e9i)T^{2} \) |
| 23 | \( 1 + (-8.20e4 - 1.42e5i)T + (-3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + (4.46e4 - 2.57e4i)T + (2.50e11 - 4.33e11i)T^{2} \) |
| 31 | \( 1 - 1.20e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 + 1.62e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + (2.26e6 + 1.30e6i)T + (3.99e12 + 6.91e12i)T^{2} \) |
| 43 | \( 1 + (-1.26e6 + 2.18e6i)T + (-5.84e12 - 1.01e13i)T^{2} \) |
| 47 | \( 1 + (4.49e6 + 7.78e6i)T + (-1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (4.96e6 - 2.86e6i)T + (3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (1.48e7 + 8.56e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (7.90e6 + 1.36e7i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-1.02e7 + 5.89e6i)T + (2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + (-2.94e7 - 1.70e7i)T + (3.22e14 + 5.59e14i)T^{2} \) |
| 73 | \( 1 + (2.15e7 - 3.73e7i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-1.76e7 - 1.01e7i)T + (7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 - 4.85e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + (1.16e7 - 6.70e6i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 + (5.17e7 + 2.98e7i)T + (3.91e15 + 6.78e15i)T^{2} \) |
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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
−14.62412921938362676909442311295, −13.85313614000506351418846485076, −12.42032706948771290711910470776, −11.21530774168975364105303609150, −9.240879942164677574078159782204, −8.519846584826939707125655035833, −6.71477109384739122206016224078, −4.96321061814212536614099519721, −3.76056808505449897311285283060, −1.73353109367369427740371772979,
1.58845535835078916720111014881, 2.81847357351695032411373592720, 4.63521202329804583563959893015, 6.47495770690369775683542372475, 7.890609483841989436357684918485, 9.406123178067115641387440711975, 10.98864577031252646783092373003, 11.98527652936184467092904387549, 13.51967921711501085028464822833, 14.40521564011800783325279645737
