L(s) = 1 | + (−0.390 − 0.920i)2-s + (−0.694 + 0.719i)4-s + (−0.999 + 0.0135i)5-s + (0.933 + 0.359i)8-s + (0.403 + 0.915i)10-s + (−0.476 − 0.879i)11-s + (0.917 + 0.396i)13-s + (−0.0339 − 0.999i)16-s + (−0.275 + 0.961i)17-s + (−0.580 + 0.814i)19-s + (0.685 − 0.728i)20-s + (−0.623 + 0.781i)22-s + (0.999 − 0.0271i)25-s + (0.00679 − 0.999i)26-s + (−0.970 + 0.242i)29-s + ⋯ |
L(s) = 1 | + (−0.390 − 0.920i)2-s + (−0.694 + 0.719i)4-s + (−0.999 + 0.0135i)5-s + (0.933 + 0.359i)8-s + (0.403 + 0.915i)10-s + (−0.476 − 0.879i)11-s + (0.917 + 0.396i)13-s + (−0.0339 − 0.999i)16-s + (−0.275 + 0.961i)17-s + (−0.580 + 0.814i)19-s + (0.685 − 0.728i)20-s + (−0.623 + 0.781i)22-s + (0.999 − 0.0271i)25-s + (0.00679 − 0.999i)26-s + (−0.970 + 0.242i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.440 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.440 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3108682049 - 0.4990781116i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3108682049 - 0.4990781116i\) |
\(L(1)\) |
\(\approx\) |
\(0.5711881352 - 0.2370705523i\) |
\(L(1)\) |
\(\approx\) |
\(0.5711881352 - 0.2370705523i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + (-0.390 - 0.920i)T \) |
| 5 | \( 1 + (-0.999 + 0.0135i)T \) |
| 11 | \( 1 + (-0.476 - 0.879i)T \) |
| 13 | \( 1 + (0.917 + 0.396i)T \) |
| 17 | \( 1 + (-0.275 + 0.961i)T \) |
| 19 | \( 1 + (-0.580 + 0.814i)T \) |
| 29 | \( 1 + (-0.970 + 0.242i)T \) |
| 31 | \( 1 + (-0.786 - 0.618i)T \) |
| 37 | \( 1 + (0.0882 - 0.996i)T \) |
| 41 | \( 1 + (-0.488 + 0.872i)T \) |
| 43 | \( 1 + (0.933 - 0.359i)T \) |
| 47 | \( 1 + (-0.826 - 0.563i)T \) |
| 53 | \( 1 + (-0.942 - 0.333i)T \) |
| 59 | \( 1 + (0.403 + 0.915i)T \) |
| 61 | \( 1 + (-0.869 - 0.494i)T \) |
| 67 | \( 1 + (0.888 - 0.458i)T \) |
| 71 | \( 1 + (-0.947 + 0.320i)T \) |
| 73 | \( 1 + (0.855 + 0.517i)T \) |
| 79 | \( 1 + (0.327 - 0.945i)T \) |
| 83 | \( 1 + (0.986 + 0.162i)T \) |
| 89 | \( 1 + (0.976 - 0.215i)T \) |
| 97 | \( 1 + (0.959 + 0.281i)T \) |
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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
−18.85620073212630529999343297563, −18.28774133791760314721603200151, −17.68543930438397023328030691336, −16.94080246557690571241544922704, −16.010871659504480017494748484307, −15.74209534420858294789295082624, −15.0938737725499339914341823585, −14.49034563211806990796389853802, −13.48151002389939702057524700473, −12.970338356937805728131811921203, −12.1332122914365130040762468727, −11.03440960416324686523535322939, −10.76866491629808395234073169758, −9.64008999956096836198706992584, −9.04455973329777810236696656586, −8.26335554197084452222929523320, −7.652211615822418619790858846348, −7.030018905713279721632973045842, −6.37622604321690412747261280666, −5.28836439945094688220233974083, −4.73283920983681356438675193726, −3.98316798336669476808206295393, −3.0328416302590410136997738786, −1.82587382520285022730501184285, −0.6847333563092311624742444887,
0.31557041730140984929581698956, 1.44065758488181231555476828843, 2.2448960622648870159342854318, 3.48730707713062019308923708180, 3.67340839442302494543831901643, 4.49415381451870097444367229984, 5.56300098482300157541170348734, 6.42856828201199761137484299047, 7.55411279449913420977120996105, 8.11282465218444551284784877818, 8.67325984776152739783499602725, 9.34550841826107240153119317881, 10.47252309856863571895989824074, 10.965819342958317306506272739409, 11.365919486126753808064637534654, 12.26381377229186598699509337676, 12.91034734236143350826653392223, 13.44421627472196649432180340126, 14.42991685226341185023046296616, 15.08335163027335513804000313756, 16.19783323691986873843426589482, 16.41207182345101618768546174777, 17.275818215117892439547774902860, 18.24184212346682687829834437387, 18.78778538308993291387506529769