L(s) = 1 | + (0.993 − 0.116i)2-s + (0.973 − 0.230i)4-s + (−0.893 − 0.448i)5-s + (−0.286 − 0.957i)7-s + (0.939 − 0.342i)8-s + (−0.939 − 0.342i)10-s + (0.0581 − 0.998i)11-s + (0.597 − 0.802i)13-s + (−0.396 − 0.918i)14-s + (0.893 − 0.448i)16-s + (−0.766 + 0.642i)17-s + (0.766 + 0.642i)19-s + (−0.973 − 0.230i)20-s + (−0.0581 − 0.998i)22-s + (0.286 − 0.957i)23-s + ⋯ |
L(s) = 1 | + (0.993 − 0.116i)2-s + (0.973 − 0.230i)4-s + (−0.893 − 0.448i)5-s + (−0.286 − 0.957i)7-s + (0.939 − 0.342i)8-s + (−0.939 − 0.342i)10-s + (0.0581 − 0.998i)11-s + (0.597 − 0.802i)13-s + (−0.396 − 0.918i)14-s + (0.893 − 0.448i)16-s + (−0.766 + 0.642i)17-s + (0.766 + 0.642i)19-s + (−0.973 − 0.230i)20-s + (−0.0581 − 0.998i)22-s + (0.286 − 0.957i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0774 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0774 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.848843710 - 1.710716560i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.848843710 - 1.710716560i\) |
\(L(1)\) |
\(\approx\) |
\(1.583719281 - 0.6451750321i\) |
\(L(1)\) |
\(\approx\) |
\(1.583719281 - 0.6451750321i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 + (0.993 - 0.116i)T \) |
| 5 | \( 1 + (-0.893 - 0.448i)T \) |
| 7 | \( 1 + (-0.286 - 0.957i)T \) |
| 11 | \( 1 + (0.0581 - 0.998i)T \) |
| 13 | \( 1 + (0.597 - 0.802i)T \) |
| 17 | \( 1 + (-0.766 + 0.642i)T \) |
| 19 | \( 1 + (0.766 + 0.642i)T \) |
| 23 | \( 1 + (0.286 - 0.957i)T \) |
| 29 | \( 1 + (-0.396 + 0.918i)T \) |
| 31 | \( 1 + (-0.686 - 0.727i)T \) |
| 37 | \( 1 + (0.173 + 0.984i)T \) |
| 41 | \( 1 + (0.993 + 0.116i)T \) |
| 43 | \( 1 + (-0.835 - 0.549i)T \) |
| 47 | \( 1 + (0.686 - 0.727i)T \) |
| 53 | \( 1 + (0.5 + 0.866i)T \) |
| 59 | \( 1 + (0.0581 + 0.998i)T \) |
| 61 | \( 1 + (0.973 + 0.230i)T \) |
| 67 | \( 1 + (0.396 + 0.918i)T \) |
| 71 | \( 1 + (0.939 + 0.342i)T \) |
| 73 | \( 1 + (-0.939 + 0.342i)T \) |
| 79 | \( 1 + (-0.993 + 0.116i)T \) |
| 83 | \( 1 + (0.993 - 0.116i)T \) |
| 89 | \( 1 + (0.939 - 0.342i)T \) |
| 97 | \( 1 + (0.893 - 0.448i)T \) |
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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
−31.122191471746341447896499556287, −30.380902076955257586495721970224, −28.91058609975304961400573301946, −28.065345253125059032246062295877, −26.49239993547263245093972098447, −25.485968285938680184901628578635, −24.41238471994531314142451460630, −23.27375352019805296186151568228, −22.52120545584238444052120869884, −21.52371624961276425253480376478, −20.21974797166535372019214424060, −19.236984273318968566179848040873, −17.89701178951059577489916719702, −16.040646448909481252404601085571, −15.5082717229088148294804748126, −14.433955702377174041962718389957, −13.05998858621189422897052326201, −11.86969916014745243039454937512, −11.19072795335627318008666467043, −9.26364489073429849872837607201, −7.5360141378229731459374079026, −6.54477574649406860485817991392, −4.99592982556053768537266030424, −3.6848892326199020627740814617, −2.29940902779348274515744147746,
0.926066688338556489881723628874, 3.30285053937033776856161020056, 4.183469049991924482277258666254, 5.71757214299450011088698620105, 7.14432227340988677431811001221, 8.40126849345515229312689265224, 10.496372917462951089714435559516, 11.36070515787223163610291844654, 12.727439742853857203694108148242, 13.54200042207455506518458823916, 14.87783891408414156263988340088, 16.062965331509978304945252309575, 16.76858781866752450559198034765, 18.80265924307911258760513030274, 20.05984723004873252870623743130, 20.50513258777255337742176360230, 22.05455928864801591720946426943, 23.02148465984601937694208590542, 23.884689410550679585418797054524, 24.70808826309106549425186089432, 26.20134496227586264374269340131, 27.32395444338298043241523725259, 28.62640008449547450216894333140, 29.62827629096155646920515331992, 30.61732160071197634959115044702