L(s) = 1 | + (0.642 − 0.766i)2-s + (0.342 − 0.939i)3-s + (−0.173 − 0.984i)4-s + (−0.5 − 0.866i)6-s + (−0.866 − 0.500i)8-s + (−0.766 − 0.642i)9-s + (−0.984 − 0.173i)12-s + (1.75 − 0.469i)13-s + (−0.939 + 0.342i)16-s + (−0.984 + 0.173i)18-s + (−0.866 − 0.5i)23-s + (−0.766 + 0.642i)24-s + (0.866 − 0.5i)25-s + (0.766 − 1.64i)26-s + (−0.866 + 0.500i)27-s + ⋯ |
L(s) = 1 | + (0.642 − 0.766i)2-s + (0.342 − 0.939i)3-s + (−0.173 − 0.984i)4-s + (−0.5 − 0.866i)6-s + (−0.866 − 0.500i)8-s + (−0.766 − 0.642i)9-s + (−0.984 − 0.173i)12-s + (1.75 − 0.469i)13-s + (−0.939 + 0.342i)16-s + (−0.984 + 0.173i)18-s + (−0.866 − 0.5i)23-s + (−0.766 + 0.642i)24-s + (0.866 − 0.5i)25-s + (0.766 − 1.64i)26-s + (−0.866 + 0.500i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.848832292\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.848832292\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.642 + 0.766i)T \) |
| 3 | \( 1 + (-0.342 + 0.939i)T \) |
| 23 | \( 1 + (0.866 + 0.5i)T \) |
good | 5 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 13 | \( 1 + (-1.75 + 0.469i)T + (0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 29 | \( 1 + (1.58 + 0.424i)T + (0.866 + 0.5i)T^{2} \) |
| 31 | \( 1 + (0.300 + 0.173i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (-0.642 + 1.11i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (-0.133 - 0.5i)T + (-0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 - 1.96iT - T^{2} \) |
| 73 | \( 1 + 0.347iT - T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−8.593340610244135828149036591961, −7.74421206397398258016104893381, −6.84371266490409358507540103687, −5.86538118408633637352293150580, −5.80913707589941868497853090166, −4.32293932424471108061268162928, −3.63183174280091759114717496966, −2.77843590368145515963471378256, −1.86437747349453085821066125432, −0.880012617110626440596186228521,
1.96460062153134222980392350556, 3.35453025517770757468060702286, 3.67814572398871947057351878949, 4.52727011643484071341612523229, 5.41031885952547068549037376134, 5.96907134625219540399774761791, 6.82094206217175262741687111516, 7.74015222028560205427804468161, 8.422330966374147753157684160012, 9.017983973505453105858593806811