L(s) = 1 | + (0.309 − 0.951i)3-s − 7-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)11-s + (−0.809 − 0.587i)13-s + (−0.309 − 0.951i)17-s + (−0.309 − 0.951i)19-s + (−0.309 + 0.951i)21-s + (0.809 − 0.587i)23-s + (−0.809 + 0.587i)27-s + (−0.309 + 0.951i)29-s + (0.309 + 0.951i)31-s + (−0.309 − 0.951i)33-s + (−0.809 − 0.587i)37-s + (−0.809 + 0.587i)39-s + ⋯ |
L(s) = 1 | + (0.309 − 0.951i)3-s − 7-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)11-s + (−0.809 − 0.587i)13-s + (−0.309 − 0.951i)17-s + (−0.309 − 0.951i)19-s + (−0.309 + 0.951i)21-s + (0.809 − 0.587i)23-s + (−0.809 + 0.587i)27-s + (−0.309 + 0.951i)29-s + (0.309 + 0.951i)31-s + (−0.309 − 0.951i)33-s + (−0.809 − 0.587i)37-s + (−0.809 + 0.587i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4606798849 - 0.8379735999i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4606798849 - 0.8379735999i\) |
\(L(1)\) |
\(\approx\) |
\(0.8380630645 - 0.4769246936i\) |
\(L(1)\) |
\(\approx\) |
\(0.8380630645 - 0.4769246936i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.309 - 0.951i)T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + (0.809 - 0.587i)T \) |
| 13 | \( 1 + (-0.809 - 0.587i)T \) |
| 17 | \( 1 + (-0.309 - 0.951i)T \) |
| 19 | \( 1 + (-0.309 - 0.951i)T \) |
| 23 | \( 1 + (0.809 - 0.587i)T \) |
| 29 | \( 1 + (-0.309 + 0.951i)T \) |
| 31 | \( 1 + (0.309 + 0.951i)T \) |
| 37 | \( 1 + (-0.809 - 0.587i)T \) |
| 41 | \( 1 + (-0.809 - 0.587i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (-0.309 + 0.951i)T \) |
| 53 | \( 1 + (0.309 - 0.951i)T \) |
| 59 | \( 1 + (0.809 + 0.587i)T \) |
| 61 | \( 1 + (0.809 - 0.587i)T \) |
| 67 | \( 1 + (0.309 + 0.951i)T \) |
| 71 | \( 1 + (0.309 - 0.951i)T \) |
| 73 | \( 1 + (0.809 - 0.587i)T \) |
| 79 | \( 1 + (0.309 - 0.951i)T \) |
| 83 | \( 1 + (0.309 + 0.951i)T \) |
| 89 | \( 1 + (-0.809 + 0.587i)T \) |
| 97 | \( 1 + (-0.309 + 0.951i)T \) |
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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
−27.146581540236877263392167100801, −26.274107403128604008315849999692, −25.536893080085484107015865310849, −24.61716559063018318793444766447, −23.1477543843218748245200411491, −22.39542309154696727405600154112, −21.62796812648621775685501032715, −20.59037390961323736065557425051, −19.57238688653047279060108563735, −19.02736818006370401812132181031, −17.11462317916465659479797723837, −16.810480442397280437383458710205, −15.4593829923634562876744817420, −14.84674485925922985049363345284, −13.72016513383307300127703926636, −12.51991374006951854849347439082, −11.42107835600971182950707286983, −10.046575400527769110662069345502, −9.5714032027999046730488842809, −8.45037151115983159539947776757, −6.99095350482237177472513450693, −5.82691498520765418851432696321, −4.38853032038681970214765221973, −3.568778187344803239033126490217, −2.145264895084835188911542033013,
0.69389153460609489405336817747, 2.49663684854194352887805900729, 3.412165703772744130978189889159, 5.22682935438834922934244502642, 6.62480938226212239112818770165, 7.133996391907092231978195959120, 8.655835014698588588953635930061, 9.37977390434927547472105245954, 10.87049136221029491914370742198, 12.0976594884517012309736627881, 12.86663337868514326116248969783, 13.79744953967671234304667022997, 14.75148520223074153234419685416, 15.998350023285272977335228645253, 17.099949634553803972934306936004, 17.99089897364909189729328011408, 19.24155789376734856979275753314, 19.55966020520372132120357428345, 20.6390720631732519244668956466, 22.160553711729395439083705616515, 22.719833212164619999131675167299, 23.92636190492385398930530567650, 24.77152412220844144464136308052, 25.44391922050916595312267324178, 26.45598364628349757858082687017