L(s) = 1 | + (−0.525 − 0.850i)2-s + (0.197 − 0.980i)3-s + (−0.448 + 0.894i)4-s + (−0.730 + 0.683i)5-s + (−0.937 + 0.346i)6-s + (−0.975 − 0.219i)7-s + (0.996 − 0.0883i)8-s + (−0.921 − 0.387i)9-s + (0.964 + 0.262i)10-s + (0.0221 − 0.999i)11-s + (0.787 + 0.616i)12-s + (−0.283 − 0.958i)13-s + (0.325 + 0.945i)14-s + (0.525 + 0.850i)15-s + (−0.598 − 0.801i)16-s + (−0.952 + 0.304i)17-s + ⋯ |
L(s) = 1 | + (−0.525 − 0.850i)2-s + (0.197 − 0.980i)3-s + (−0.448 + 0.894i)4-s + (−0.730 + 0.683i)5-s + (−0.937 + 0.346i)6-s + (−0.975 − 0.219i)7-s + (0.996 − 0.0883i)8-s + (−0.921 − 0.387i)9-s + (0.964 + 0.262i)10-s + (0.0221 − 0.999i)11-s + (0.787 + 0.616i)12-s + (−0.283 − 0.958i)13-s + (0.325 + 0.945i)14-s + (0.525 + 0.850i)15-s + (−0.598 − 0.801i)16-s + (−0.952 + 0.304i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1731067694 + 0.05868659629i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1731067694 + 0.05868659629i\) |
\(L(1)\) |
\(\approx\) |
\(0.4306660114 - 0.2838887111i\) |
\(L(1)\) |
\(\approx\) |
\(0.4306660114 - 0.2838887111i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 569 | \( 1 \) |
good | 2 | \( 1 + (-0.525 - 0.850i)T \) |
| 3 | \( 1 + (0.197 - 0.980i)T \) |
| 5 | \( 1 + (-0.730 + 0.683i)T \) |
| 7 | \( 1 + (-0.975 - 0.219i)T \) |
| 11 | \( 1 + (0.0221 - 0.999i)T \) |
| 13 | \( 1 + (-0.283 - 0.958i)T \) |
| 17 | \( 1 + (-0.952 + 0.304i)T \) |
| 19 | \( 1 + (0.448 + 0.894i)T \) |
| 23 | \( 1 + (-0.408 - 0.912i)T \) |
| 29 | \( 1 + (0.952 + 0.304i)T \) |
| 31 | \( 1 + (-0.984 - 0.176i)T \) |
| 37 | \( 1 + (0.975 + 0.219i)T \) |
| 41 | \( 1 + (0.408 + 0.912i)T \) |
| 43 | \( 1 + (-0.525 + 0.850i)T \) |
| 47 | \( 1 + (-0.154 + 0.988i)T \) |
| 53 | \( 1 + (-0.937 + 0.346i)T \) |
| 59 | \( 1 + (-0.154 + 0.988i)T \) |
| 61 | \( 1 + (0.633 + 0.773i)T \) |
| 67 | \( 1 + (-0.952 - 0.304i)T \) |
| 71 | \( 1 + (0.996 + 0.0883i)T \) |
| 73 | \( 1 + (0.448 + 0.894i)T \) |
| 79 | \( 1 + (0.759 + 0.650i)T \) |
| 83 | \( 1 + (-0.903 - 0.428i)T \) |
| 89 | \( 1 + (-0.984 + 0.176i)T \) |
| 97 | \( 1 + (-0.0663 + 0.997i)T \) |
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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
−23.303001864617379174887016601505, −22.415974539342974212735662154218, −21.73403297511087857441562844196, −20.307179056426810740812748630, −19.818716442865359347775631235476, −19.21841949473093613387261192138, −17.92416520502961892188000717672, −17.04188127335755309380939175839, −16.26017835255071692836975006184, −15.65431904955713343882408623933, −15.23836510159808540077772317473, −14.08531863577018163604652414065, −13.16485978549407142401031689743, −11.941227497096400012218019778348, −10.98541542406023104856748770174, −9.682728594437905608017599478192, −9.37267597959152754355187334377, −8.602498420883256152164841639335, −7.438086225616815573024830106998, −6.62665132880111406310851228979, −5.29369102788501220862496982782, −4.59094194417145592839596376622, −3.73848229691945342116015856092, −2.138775878339233175529190798870, −0.129215143816941581353281521387,
1.050106771974739236484709983310, 2.69207582831069692190691551258, 3.085097135739617619073262480772, 4.12787954306191801681353083052, 6.00343359327106536693978770487, 6.85567020323793031816720379279, 7.88987752381930377034188773015, 8.37408766833308207533834128876, 9.581795828267306095598547796097, 10.63324337209393243524141947782, 11.28785946267788274967802677371, 12.32023806201190784724390567492, 12.872829578648621551083895111216, 13.75735811495649016339908412217, 14.68399127227441407598470505208, 16.02836611261658021388966886806, 16.74188830560614774949771188107, 18.010424728756346407288392538078, 18.40202650683186141632509149971, 19.32030077610210333336246013647, 19.76893790276753039570796230867, 20.37321631911428554936047390377, 21.83167298994814262826872437069, 22.52416452419983214805318462039, 23.09766463808722991305965036960