L(s) = 1 | + (0.797 + 0.603i)2-s + (0.272 + 0.962i)4-s + (0.710 + 0.703i)5-s + (0.179 + 0.983i)7-s + (−0.362 + 0.931i)8-s + (0.142 + 0.989i)10-s + (0.830 − 0.556i)13-s + (−0.449 + 0.893i)14-s + (−0.851 + 0.524i)16-s + (−0.736 + 0.676i)17-s + (−0.198 − 0.980i)19-s + (−0.483 + 0.875i)20-s + (−0.723 + 0.690i)23-s + (0.00951 + 0.999i)25-s + (0.998 + 0.0570i)26-s + ⋯ |
L(s) = 1 | + (0.797 + 0.603i)2-s + (0.272 + 0.962i)4-s + (0.710 + 0.703i)5-s + (0.179 + 0.983i)7-s + (−0.362 + 0.931i)8-s + (0.142 + 0.989i)10-s + (0.830 − 0.556i)13-s + (−0.449 + 0.893i)14-s + (−0.851 + 0.524i)16-s + (−0.736 + 0.676i)17-s + (−0.198 − 0.980i)19-s + (−0.483 + 0.875i)20-s + (−0.723 + 0.690i)23-s + (0.00951 + 0.999i)25-s + (0.998 + 0.0570i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.799 + 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.799 + 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8512057315 + 2.549484083i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8512057315 + 2.549484083i\) |
\(L(1)\) |
\(\approx\) |
\(1.370434189 + 1.200363469i\) |
\(L(1)\) |
\(\approx\) |
\(1.370434189 + 1.200363469i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.797 + 0.603i)T \) |
| 5 | \( 1 + (0.710 + 0.703i)T \) |
| 7 | \( 1 + (0.179 + 0.983i)T \) |
| 13 | \( 1 + (0.830 - 0.556i)T \) |
| 17 | \( 1 + (-0.736 + 0.676i)T \) |
| 19 | \( 1 + (-0.198 - 0.980i)T \) |
| 23 | \( 1 + (-0.723 + 0.690i)T \) |
| 29 | \( 1 + (0.861 + 0.508i)T \) |
| 31 | \( 1 + (0.640 - 0.768i)T \) |
| 37 | \( 1 + (-0.466 + 0.884i)T \) |
| 41 | \( 1 + (-0.683 + 0.730i)T \) |
| 43 | \( 1 + (0.888 + 0.458i)T \) |
| 47 | \( 1 + (0.905 - 0.424i)T \) |
| 53 | \( 1 + (0.0285 - 0.999i)T \) |
| 59 | \( 1 + (0.290 - 0.956i)T \) |
| 61 | \( 1 + (-0.123 - 0.992i)T \) |
| 67 | \( 1 + (0.981 - 0.189i)T \) |
| 71 | \( 1 + (-0.993 - 0.113i)T \) |
| 73 | \( 1 + (-0.610 - 0.791i)T \) |
| 79 | \( 1 + (0.935 + 0.353i)T \) |
| 83 | \( 1 + (-0.432 + 0.901i)T \) |
| 89 | \( 1 + (-0.415 - 0.909i)T \) |
| 97 | \( 1 + (-0.710 + 0.703i)T \) |
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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
−20.898031172422978886103649280314, −20.652324130758334388804794240665, −19.88751286995864575852676665415, −18.980169137695719002029651380312, −18.05708737623863327832474531495, −17.27754463795900304393645516008, −16.24127046625780782633552925317, −15.80614399214771813468205166792, −14.40581293706963042220972127222, −13.85042551331594557318832728800, −13.472368257246599057453874939844, −12.4309590666320122226510300078, −11.83016058904430286245859608384, −10.64104866440833384884504213065, −10.30704465664082051066443365894, −9.245152950241762073572807014962, −8.441197806990214646803052762845, −7.08560742330860390064459639047, −6.26473736047633866210082009148, −5.48115388609998853805444709926, −4.349556694371723473173780511804, −4.064376183420195919038960846764, −2.639937398573495850123227294786, −1.69166441369533080981153870403, −0.83623854998722449515059663872,
1.82715336891455514484106353294, 2.66947125037846788604254417826, 3.46440948144925186122002358465, 4.658128142843998610068836469587, 5.561264104826082598889210236875, 6.23899457258733902164475650172, 6.81163382567072720247369250448, 8.07516801973161712324170117880, 8.6680668681127305937416365247, 9.71404477322868501022038239987, 10.882091376286045174942621744790, 11.48236581880649089159787019433, 12.484675323289076182994029728315, 13.32156821665392007452344522541, 13.82799598748821570731195128927, 14.83552196842336669726359606878, 15.39314726949378414739711404173, 15.921614917349161444912853784118, 17.23460538430851582851285127142, 17.73239184924688241363239864149, 18.36424734015598915219246076309, 19.41312676872418108434770888905, 20.48878434985847191828800623831, 21.27782172989361523260462416348, 21.99744994110104768363238701375