L(s) = 1 | + (0.991 − 0.130i)2-s + (0.130 + 0.991i)3-s + (0.965 − 0.258i)4-s + (0.258 + 0.965i)6-s + (0.923 − 0.382i)8-s + (−0.965 + 0.258i)9-s + (1.24 − 0.423i)11-s + (0.382 + 0.923i)12-s + (0.866 − 0.5i)16-s + (−0.793 + 0.608i)17-s + (−0.923 + 0.382i)18-s + (−1.78 − 0.739i)19-s + (1.18 − 0.583i)22-s + (0.5 + 0.866i)24-s + (−0.608 + 0.793i)25-s + ⋯ |
L(s) = 1 | + (0.991 − 0.130i)2-s + (0.130 + 0.991i)3-s + (0.965 − 0.258i)4-s + (0.258 + 0.965i)6-s + (0.923 − 0.382i)8-s + (−0.965 + 0.258i)9-s + (1.24 − 0.423i)11-s + (0.382 + 0.923i)12-s + (0.866 − 0.5i)16-s + (−0.793 + 0.608i)17-s + (−0.923 + 0.382i)18-s + (−1.78 − 0.739i)19-s + (1.18 − 0.583i)22-s + (0.5 + 0.866i)24-s + (−0.608 + 0.793i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.007807514\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.007807514\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.991 + 0.130i)T \) |
| 3 | \( 1 + (-0.130 - 0.991i)T \) |
| 17 | \( 1 + (0.793 - 0.608i)T \) |
good | 5 | \( 1 + (0.608 - 0.793i)T^{2} \) |
| 7 | \( 1 + (-0.608 - 0.793i)T^{2} \) |
| 11 | \( 1 + (-1.24 + 0.423i)T + (0.793 - 0.608i)T^{2} \) |
| 13 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 19 | \( 1 + (1.78 + 0.739i)T + (0.707 + 0.707i)T^{2} \) |
| 23 | \( 1 + (0.130 + 0.991i)T^{2} \) |
| 29 | \( 1 + (0.991 + 0.130i)T^{2} \) |
| 31 | \( 1 + (0.793 + 0.608i)T^{2} \) |
| 37 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 41 | \( 1 + (-1.50 + 0.0983i)T + (0.991 - 0.130i)T^{2} \) |
| 43 | \( 1 + (1.25 + 0.965i)T + (0.258 + 0.965i)T^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 59 | \( 1 + (0.258 - 1.96i)T + (-0.965 - 0.258i)T^{2} \) |
| 61 | \( 1 + (0.608 + 0.793i)T^{2} \) |
| 67 | \( 1 + (1.05 + 0.608i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 73 | \( 1 + (1.57 + 1.05i)T + (0.382 + 0.923i)T^{2} \) |
| 79 | \( 1 + (0.793 - 0.608i)T^{2} \) |
| 83 | \( 1 + (0.241 + 1.83i)T + (-0.965 + 0.258i)T^{2} \) |
| 89 | \( 1 + (0.541 - 0.541i)T - iT^{2} \) |
| 97 | \( 1 + (-0.130 - 0.00855i)T + (0.991 + 0.130i)T^{2} \) |
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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
−10.26570928214288092219450048955, −9.111688876922513650795179574829, −8.656533936875713891654615741936, −7.34857998577127559776121147628, −6.30345749177852690716959207972, −5.80567055211901517020448968073, −4.47867029548697079139962247001, −4.15509496773469096406926599755, −3.14050757326945326487550271382, −1.96214863401798239993717702003,
1.68873102546366555531317735430, 2.53179617526972211247145990739, 3.83676730353034122871704501253, 4.59175167398295220337417372045, 5.92010852753724758540624409456, 6.48246365486326889639212447112, 7.07491683827259980883275317079, 8.063493045917027443104086942020, 8.759897735693884519056334030822, 9.894260875815049395855754641282