| L(s) = 1 | + (1.12 − 0.861i)2-s + (0.684 − 1.59i)3-s + (0.513 − 1.93i)4-s + (−1.15 + 3.17i)5-s + (−0.603 − 2.37i)6-s + (0.593 − 3.36i)7-s + (−1.08 − 2.61i)8-s + (−2.06 − 2.17i)9-s + (1.43 + 4.55i)10-s + (−0.197 − 0.541i)11-s + (−2.72 − 2.14i)12-s + (4.30 + 5.12i)13-s + (−2.23 − 4.28i)14-s + (4.25 + 4.00i)15-s + (−3.47 − 1.98i)16-s + (1.15 + 2.00i)17-s + ⋯ |
| L(s) = 1 | + (0.792 − 0.609i)2-s + (0.395 − 0.918i)3-s + (0.256 − 0.966i)4-s + (−0.516 + 1.41i)5-s + (−0.246 − 0.969i)6-s + (0.224 − 1.27i)7-s + (−0.385 − 0.922i)8-s + (−0.687 − 0.726i)9-s + (0.455 + 1.43i)10-s + (−0.0594 − 0.163i)11-s + (−0.786 − 0.618i)12-s + (1.19 + 1.42i)13-s + (−0.597 − 1.14i)14-s + (1.09 + 1.03i)15-s + (−0.867 − 0.496i)16-s + (0.281 + 0.486i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00917 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00917 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.33727 - 1.34959i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33727 - 1.34959i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.12 + 0.861i)T \) |
| 3 | \( 1 + (-0.684 + 1.59i)T \) |
| good | 5 | \( 1 + (1.15 - 3.17i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.593 + 3.36i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (0.197 + 0.541i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-4.30 - 5.12i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.15 - 2.00i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.353 + 0.203i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.15 - 6.53i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (2.24 - 2.67i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.382 + 2.17i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (1.05 - 0.607i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.09 + 4.27i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.442 - 1.21i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.547 - 3.10i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 9.35iT - 53T^{2} \) |
| 59 | \( 1 + (3.00 - 8.25i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (4.67 + 0.824i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (9.67 + 11.5i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (3.92 + 6.79i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.641 + 1.11i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.84 - 3.22i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-6.67 + 7.95i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (2.86 - 4.96i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.5 + 3.84i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.96256888849444238927982074347, −11.15012789885211613040367469753, −10.70606240046383526603174798688, −9.265361189711232109886823385277, −7.65019699747907497337965787478, −6.92121959771620367670437128778, −6.10193303073939935580545819837, −3.99233873277079265481285051892, −3.30383169742771505828829195713, −1.61535556040296628229038898140,
2.86714277646663757930746351609, 4.21284549893604261536876860384, 5.17192633739653517759487775480, 5.84500601679552968864764001296, 7.948669219836014542485653954262, 8.496983831460567053210706581802, 9.165257662222716600297175546402, 10.80418536311644969205430648450, 11.90934352029209750239651090072, 12.64242854052321167065227625806
