L(s) = 1 | + (0.555 − 0.831i)3-s + (0.195 − 0.980i)5-s + (−0.382 − 0.923i)9-s + (−0.831 + 0.555i)11-s + (−0.195 − 0.980i)13-s + (−0.707 − 0.707i)15-s + (−0.707 + 0.707i)17-s + (−0.980 + 0.195i)19-s + (−0.923 + 0.382i)23-s + (−0.923 − 0.382i)25-s + (−0.980 − 0.195i)27-s + (−0.831 − 0.555i)29-s − i·31-s − i·33-s + (0.980 + 0.195i)37-s + ⋯ |
L(s) = 1 | + (0.555 − 0.831i)3-s + (0.195 − 0.980i)5-s + (−0.382 − 0.923i)9-s + (−0.831 + 0.555i)11-s + (−0.195 − 0.980i)13-s + (−0.707 − 0.707i)15-s + (−0.707 + 0.707i)17-s + (−0.980 + 0.195i)19-s + (−0.923 + 0.382i)23-s + (−0.923 − 0.382i)25-s + (−0.980 − 0.195i)27-s + (−0.831 − 0.555i)29-s − i·31-s − i·33-s + (0.980 + 0.195i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1277384368 - 0.7361747511i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1277384368 - 0.7361747511i\) |
\(L(1)\) |
\(\approx\) |
\(0.8002080043 - 0.5118993207i\) |
\(L(1)\) |
\(\approx\) |
\(0.8002080043 - 0.5118993207i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.555 - 0.831i)T \) |
| 5 | \( 1 + (0.195 - 0.980i)T \) |
| 11 | \( 1 + (-0.831 + 0.555i)T \) |
| 13 | \( 1 + (-0.195 - 0.980i)T \) |
| 17 | \( 1 + (-0.707 + 0.707i)T \) |
| 19 | \( 1 + (-0.980 + 0.195i)T \) |
| 23 | \( 1 + (-0.923 + 0.382i)T \) |
| 29 | \( 1 + (-0.831 - 0.555i)T \) |
| 31 | \( 1 - iT \) |
| 37 | \( 1 + (0.980 + 0.195i)T \) |
| 41 | \( 1 + (0.923 - 0.382i)T \) |
| 43 | \( 1 + (-0.555 - 0.831i)T \) |
| 47 | \( 1 + (-0.707 + 0.707i)T \) |
| 53 | \( 1 + (-0.831 + 0.555i)T \) |
| 59 | \( 1 + (0.195 - 0.980i)T \) |
| 61 | \( 1 + (0.555 - 0.831i)T \) |
| 67 | \( 1 + (0.555 - 0.831i)T \) |
| 71 | \( 1 + (0.382 - 0.923i)T \) |
| 73 | \( 1 + (-0.382 - 0.923i)T \) |
| 79 | \( 1 + (0.707 + 0.707i)T \) |
| 83 | \( 1 + (0.980 - 0.195i)T \) |
| 89 | \( 1 + (-0.923 - 0.382i)T \) |
| 97 | \( 1 - iT \) |
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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
−22.12795811494385249408900980917, −21.650339814334327426291742290822, −20.96527965705044478671552496691, −20.06371453516908980078817525718, −19.20667756567968001618881242253, −18.56292922990086962106285708976, −17.74017707937181613004389497796, −16.524702134657713040524479779012, −16.07591955434708577031101937624, −15.00305872353647256768348044366, −14.57479171520431541801689067226, −13.68705998612500883105837424811, −13.045119764073661192707702714253, −11.39992498304555447326816055989, −11.09139972965495718080686657691, −10.09298989231839225092367018535, −9.460355785661615308420259253350, −8.49226326859413528028250266328, −7.62900416169535569756668369171, −6.62270856068483412894386268344, −5.687307028619130179838937736632, −4.551525750837032570330007635235, −3.78713744239796069564659906380, −2.628934392665722452363434653165, −2.18791627846611808604476299618,
0.27056094231487583971884017573, 1.69886092156635508059863640018, 2.32398086338710674061806416897, 3.60713585219508805027149149177, 4.670355172091743783934865607066, 5.69665084158786727902852965491, 6.49544073345298418409805791294, 7.85388820670588368729344791218, 8.05494729563480725104610113367, 9.08915432503943287205131563382, 9.92014589391721865016004619175, 10.94439906764922190057364277447, 12.20483564791134460469111780552, 12.80167689883023926253071189347, 13.17569662598830849531049024332, 14.17423759099121110871693431857, 15.173619168768878712149750932455, 15.7426188683186607466034568066, 16.99678769677466584279351453314, 17.62540413407961306715876045297, 18.23162839106182315002333622756, 19.33160733174897391037728225050, 19.94777355082706356182280087726, 20.57560636412361053153041708568, 21.26613057075355110765788939053