| L(s) = 1 | + (0.939 + 0.342i)2-s + (0.102 + 0.581i)3-s + (0.766 + 0.642i)4-s + (−1.23 + 1.03i)5-s + (−0.102 + 0.581i)6-s + (−1.96 + 3.40i)7-s + (0.500 + 0.866i)8-s + (2.49 − 0.906i)9-s + (−1.51 + 0.551i)10-s + (−0.5 − 0.866i)11-s + (−0.295 + 0.511i)12-s + (−0.459 + 2.60i)13-s + (−3.01 + 2.52i)14-s + (−0.729 − 0.611i)15-s + (0.173 + 0.984i)16-s + (−7.29 − 2.65i)17-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.241i)2-s + (0.0592 + 0.335i)3-s + (0.383 + 0.321i)4-s + (−0.552 + 0.463i)5-s + (−0.0418 + 0.237i)6-s + (−0.743 + 1.28i)7-s + (0.176 + 0.306i)8-s + (0.830 − 0.302i)9-s + (−0.478 + 0.174i)10-s + (−0.150 − 0.261i)11-s + (−0.0852 + 0.147i)12-s + (−0.127 + 0.723i)13-s + (−0.805 + 0.676i)14-s + (−0.188 − 0.157i)15-s + (0.0434 + 0.246i)16-s + (−1.76 − 0.643i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.312 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.312 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.998522 + 1.37965i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.998522 + 1.37965i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 - 0.342i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (-4.10 - 1.45i)T \) |
| good | 3 | \( 1 + (-0.102 - 0.581i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (1.23 - 1.03i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (1.96 - 3.40i)T + (-3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (0.459 - 2.60i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (7.29 + 2.65i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-3.86 - 3.24i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-2.18 + 0.795i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-1.24 + 2.16i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 4.93T + 37T^{2} \) |
| 41 | \( 1 + (0.540 + 3.06i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-8.46 + 7.10i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-7.71 + 2.80i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-3.54 - 2.97i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (3.61 + 1.31i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (10.1 + 8.47i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (8.96 - 3.26i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-7.60 + 6.38i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-0.641 - 3.64i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-0.851 - 4.82i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (1.50 - 2.61i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.00 + 5.71i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-10.1 - 3.70i)T + (74.3 + 62.3i)T^{2} \) |
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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
−11.63789459572829992001013965135, −10.79754730745560383040406169653, −9.409221683265676997192070781105, −9.006753427958748172225442108801, −7.46474124657869033586257798476, −6.76826885195923669991613982600, −5.75386622671856457796328035099, −4.59499953589672566474650162253, −3.53337990374082942607979647818, −2.46549859067660640697115461429,
0.923629285013996521701303164004, 2.77224810094656189347428711503, 4.19093336402293182962971797770, 4.65679015083447387775199405045, 6.32628925792666936250995940606, 7.13660180007868946677314560806, 7.88120230595193166134381700625, 9.223206408190071833023000547902, 10.38595650908736766900925135585, 10.80533009881160774633280459765
