| 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
−11.58430027353199517046125112721, −11.23098577191901157171288922280, −10.25672473204779234426577578729, −9.237448765305623963630286468426, −7.997550351092941319422433119645, −7.04065730231961848225160952291, −6.09473708022961811500504692496, −4.38359362957214569965582547287, −2.56912038333181470744147345184, −1.17303552817704516764855195887,
1.70016536622811754048445052805, 4.51235275141365586147731899202, 5.46360940331094399406151088529, 5.99274789073743377707924426891, 7.73940147734872935521249516301, 8.608369090387386595660663183330, 9.552225274321500454357324942864, 10.45258009355795780273867352782, 11.09909725080829361268344295921, 12.42523132675562820360394689038
