| L(s) = 1 | + (0.906 − 0.422i)5-s + (0.422 − 0.906i)11-s + (−0.573 + 0.819i)13-s − 17-s + (0.707 + 0.707i)19-s + (0.984 − 0.173i)23-s + (0.642 − 0.766i)25-s + (−0.819 + 0.573i)29-s + (−0.766 + 0.642i)31-s + (−0.258 − 0.965i)37-s + (−0.984 + 0.173i)41-s + (0.996 − 0.0871i)43-s + (−0.766 − 0.642i)47-s + (−0.258 − 0.965i)53-s − i·55-s + ⋯ |
| L(s) = 1 | + (0.906 − 0.422i)5-s + (0.422 − 0.906i)11-s + (−0.573 + 0.819i)13-s − 17-s + (0.707 + 0.707i)19-s + (0.984 − 0.173i)23-s + (0.642 − 0.766i)25-s + (−0.819 + 0.573i)29-s + (−0.766 + 0.642i)31-s + (−0.258 − 0.965i)37-s + (−0.984 + 0.173i)41-s + (0.996 − 0.0871i)43-s + (−0.766 − 0.642i)47-s + (−0.258 − 0.965i)53-s − i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0902 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0902 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.106013960 - 1.210792661i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.106013960 - 1.210792661i\) |
| \(L(1)\) |
\(\approx\) |
\(1.136395838 - 0.2137796231i\) |
| \(L(1)\) |
\(\approx\) |
\(1.136395838 - 0.2137796231i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (0.906 - 0.422i)T \) |
| 11 | \( 1 + (0.422 - 0.906i)T \) |
| 13 | \( 1 + (-0.573 + 0.819i)T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + (0.707 + 0.707i)T \) |
| 23 | \( 1 + (0.984 - 0.173i)T \) |
| 29 | \( 1 + (-0.819 + 0.573i)T \) |
| 31 | \( 1 + (-0.766 + 0.642i)T \) |
| 37 | \( 1 + (-0.258 - 0.965i)T \) |
| 41 | \( 1 + (-0.984 + 0.173i)T \) |
| 43 | \( 1 + (0.996 - 0.0871i)T \) |
| 47 | \( 1 + (-0.766 - 0.642i)T \) |
| 53 | \( 1 + (-0.258 - 0.965i)T \) |
| 59 | \( 1 + (-0.819 - 0.573i)T \) |
| 61 | \( 1 + (-0.0871 - 0.996i)T \) |
| 67 | \( 1 + (-0.422 - 0.906i)T \) |
| 71 | \( 1 + (0.866 - 0.5i)T \) |
| 73 | \( 1 + (-0.866 + 0.5i)T \) |
| 79 | \( 1 + (0.939 - 0.342i)T \) |
| 83 | \( 1 + (-0.819 + 0.573i)T \) |
| 89 | \( 1 - iT \) |
| 97 | \( 1 + (-0.766 - 0.642i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.87494516418941927188501694757, −17.166994595127877536586104250, −16.94864135051999826626697078273, −15.72149885432596687725350374653, −15.1209160967509677072276638070, −14.78195877506875707503135802559, −13.874674901016752177971315852378, −13.23842948719030983034014554884, −12.83637600416211679118488183778, −11.9193169543235159844053664805, −11.19149163505384377881595690202, −10.59021352478317942119469682924, −9.79500356735881934763297532182, −9.36978348824687269320804151056, −8.74293012572584169886006604163, −7.53319321087456860769772698876, −7.19937593588781543953775518195, −6.43474774020927245933028539980, −5.67625855625141761407557010068, −4.98855029998998254080200054324, −4.34015761591883415453877648105, −3.23749725345580279129036744703, −2.61163834136871852129229587691, −1.9117088559858676228289237001, −1.05043237167154441624421865686,
0.4008740516890376882702274056, 1.59788367921287367393792668158, 1.949764563511846499383473363931, 3.06540984395170215635968356330, 3.72552594954945826415407842903, 4.75608096334397647985574686422, 5.24476476557343421347568917952, 6.02802595679457057471114917289, 6.70082619989075947133411334235, 7.31184452019058338138250223653, 8.36808417736288934105058833780, 9.05020629802060112048017736306, 9.32890039123124468876829835197, 10.18239622185535899308208475926, 11.01551012661540310674101648672, 11.457538055356651507859879634905, 12.46470877722430772823089916226, 12.86067888567165446072089247503, 13.75724038256185242165607969481, 14.13191252919624128958317415227, 14.732468152424894236260617990180, 15.672124435956164889267969282972, 16.541274363619524203839687175281, 16.68982483148836683519004536266, 17.468403177074906189092460055844
