L(s) = 1 | + (−0.0203 + 0.999i)2-s + (−0.999 − 0.0407i)4-s + (0.339 − 0.940i)5-s + (0.0611 − 0.998i)8-s + (0.933 + 0.359i)10-s + (−0.768 + 0.639i)11-s + (0.557 − 0.830i)13-s + (0.996 + 0.0815i)16-s + (−0.999 + 0.0407i)17-s + (0.654 − 0.755i)19-s + (−0.377 + 0.925i)20-s + (−0.623 − 0.781i)22-s + (−0.768 − 0.639i)25-s + (0.818 + 0.574i)26-s + (0.999 − 0.0407i)29-s + ⋯ |
L(s) = 1 | + (−0.0203 + 0.999i)2-s + (−0.999 − 0.0407i)4-s + (0.339 − 0.940i)5-s + (0.0611 − 0.998i)8-s + (0.933 + 0.359i)10-s + (−0.768 + 0.639i)11-s + (0.557 − 0.830i)13-s + (0.996 + 0.0815i)16-s + (−0.999 + 0.0407i)17-s + (0.654 − 0.755i)19-s + (−0.377 + 0.925i)20-s + (−0.623 − 0.781i)22-s + (−0.768 − 0.639i)25-s + (0.818 + 0.574i)26-s + (0.999 − 0.0407i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.027844853 - 0.4522952407i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.027844853 - 0.4522952407i\) |
\(L(1)\) |
\(\approx\) |
\(0.9073048184 + 0.1641763235i\) |
\(L(1)\) |
\(\approx\) |
\(0.9073048184 + 0.1641763235i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + (-0.0203 + 0.999i)T \) |
| 5 | \( 1 + (0.339 - 0.940i)T \) |
| 11 | \( 1 + (-0.768 + 0.639i)T \) |
| 13 | \( 1 + (0.557 - 0.830i)T \) |
| 17 | \( 1 + (-0.999 + 0.0407i)T \) |
| 19 | \( 1 + (0.654 - 0.755i)T \) |
| 29 | \( 1 + (0.999 - 0.0407i)T \) |
| 31 | \( 1 + (-0.959 - 0.281i)T \) |
| 37 | \( 1 + (-0.101 + 0.994i)T \) |
| 41 | \( 1 + (-0.339 + 0.940i)T \) |
| 43 | \( 1 + (0.0611 + 0.998i)T \) |
| 47 | \( 1 + (0.900 + 0.433i)T \) |
| 53 | \( 1 + (0.685 - 0.728i)T \) |
| 59 | \( 1 + (0.933 + 0.359i)T \) |
| 61 | \( 1 + (0.818 - 0.574i)T \) |
| 67 | \( 1 + (-0.415 - 0.909i)T \) |
| 71 | \( 1 + (0.452 - 0.891i)T \) |
| 73 | \( 1 + (0.262 - 0.965i)T \) |
| 79 | \( 1 + (0.142 - 0.989i)T \) |
| 83 | \( 1 + (-0.523 + 0.852i)T \) |
| 89 | \( 1 + (0.742 - 0.670i)T \) |
| 97 | \( 1 + (-0.841 - 0.540i)T \) |
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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
−18.85745730258803796744637484272, −18.36150573507557257218974333556, −17.87497080049326675220927801485, −17.06773126849338042482528252727, −16.112761640360425291034823403195, −15.46469708438658919216492713830, −14.35256228019788480374343136906, −13.99345325580583677666241029759, −13.41062364811066201241989343969, −12.629696006672199980186700221173, −11.69438918910125037465777678295, −11.206162461110662335247249929975, −10.47401554493310606249710143318, −10.10436406830801236181494367774, −8.99456225110449886676948081649, −8.634968290284091370626875862001, −7.53624650972476268827491141206, −6.79948239972938688733128350153, −5.76568078522445513468528332480, −5.27148790796254281267654707922, −3.98264445855683338838839885594, −3.61710002783099070238663527462, −2.527244106894152142118593437884, −2.141028310119913383915953328821, −1.014221183894514202063247242985,
0.389811156705224941875996214931, 1.34273056904833315027431372750, 2.515964701665608921985488762004, 3.577035625259708650350258772797, 4.71337335355174951453350580888, 4.90282692142625834287366788018, 5.79814155245072710368187455561, 6.49379308056789605104824051993, 7.38162441081128265324999016988, 8.11007181294718234319299395715, 8.656388849164611070127103830889, 9.424968674208165509379705821017, 10.02591731137947186412560746428, 10.87046607500111988118911011761, 11.95451821445183776730197164920, 12.8453550651895218314925089528, 13.23766005386455378941413298566, 13.7169641102574202508988844842, 14.77053715074657739984691101060, 15.4933897092740046507582781678, 15.87158901057661091197440536878, 16.58502619148359477147257582642, 17.372233627567759479142314565489, 18.04016139256373388329504621906, 18.16415012507707976185797446987