| L(s) = 1 | + (−0.940 + 1.05i)2-s + (−1.04 + 1.36i)3-s + (−0.229 − 1.98i)4-s + (1.80 − 1.38i)5-s + (−0.454 − 2.38i)6-s + (2.29 + 1.30i)7-s + (2.31 + 1.62i)8-s + (0.0139 + 0.0522i)9-s + (−0.235 + 3.20i)10-s + (−0.940 + 0.123i)11-s + (2.94 + 1.76i)12-s + (2.44 − 1.01i)13-s + (−3.54 + 1.19i)14-s + 3.89i·15-s + (−3.89 + 0.911i)16-s + (0.0434 − 0.0251i)17-s + ⋯ |
| L(s) = 1 | + (−0.665 + 0.746i)2-s + (−0.603 + 0.786i)3-s + (−0.114 − 0.993i)4-s + (0.805 − 0.617i)5-s + (−0.185 − 0.973i)6-s + (0.868 + 0.495i)7-s + (0.817 + 0.575i)8-s + (0.00466 + 0.0174i)9-s + (−0.0744 + 1.01i)10-s + (−0.283 + 0.0373i)11-s + (0.850 + 0.509i)12-s + (0.677 − 0.280i)13-s + (−0.947 + 0.319i)14-s + 1.00i·15-s + (−0.973 + 0.227i)16-s + (0.0105 − 0.00608i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0291 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0291 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.619805 + 0.638166i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.619805 + 0.638166i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.940 - 1.05i)T \) |
| 7 | \( 1 + (-2.29 - 1.30i)T \) |
| good | 3 | \( 1 + (1.04 - 1.36i)T + (-0.776 - 2.89i)T^{2} \) |
| 5 | \( 1 + (-1.80 + 1.38i)T + (1.29 - 4.82i)T^{2} \) |
| 11 | \( 1 + (0.940 - 0.123i)T + (10.6 - 2.84i)T^{2} \) |
| 13 | \( 1 + (-2.44 + 1.01i)T + (9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + (-0.0434 + 0.0251i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.771 - 5.86i)T + (-18.3 - 4.91i)T^{2} \) |
| 23 | \( 1 + (0.0591 + 0.220i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (0.876 + 2.11i)T + (-20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + (-4.67 - 8.08i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.71 + 3.62i)T + (9.57 - 35.7i)T^{2} \) |
| 41 | \( 1 + (-1.81 + 1.81i)T - 41iT^{2} \) |
| 43 | \( 1 + (-3.88 + 9.36i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (-5.15 - 2.97i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (12.4 - 1.63i)T + (51.1 - 13.7i)T^{2} \) |
| 59 | \( 1 + (1.13 + 8.65i)T + (-56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (1.78 + 0.235i)T + (58.9 + 15.7i)T^{2} \) |
| 67 | \( 1 + (-4.75 + 6.19i)T + (-17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (7.65 + 7.65i)T + 71iT^{2} \) |
| 73 | \( 1 + (-0.769 - 0.206i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (10.5 + 6.10i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (13.0 - 5.39i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (2.08 - 0.558i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
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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
−12.42523132675562820360394689038, −11.09909725080829361268344295921, −10.45258009355795780273867352782, −9.552225274321500454357324942864, −8.608369090387386595660663183330, −7.73940147734872935521249516301, −5.99274789073743377707924426891, −5.46360940331094399406151088529, −4.51235275141365586147731899202, −1.70016536622811754048445052805,
1.17303552817704516764855195887, 2.56912038333181470744147345184, 4.38359362957214569965582547287, 6.09473708022961811500504692496, 7.04065730231961848225160952291, 7.997550351092941319422433119645, 9.237448765305623963630286468426, 10.25672473204779234426577578729, 11.23098577191901157171288922280, 11.58430027353199517046125112721
