L(s) = 1 | + (−0.900 − 0.433i)2-s + (0.623 − 0.781i)3-s + (0.623 + 0.781i)4-s + (−0.900 − 0.433i)5-s + (−0.900 + 0.433i)6-s + (0.623 − 0.781i)7-s + (−0.222 − 0.974i)8-s + (−0.222 − 0.974i)9-s + (0.623 + 0.781i)10-s + (−0.222 + 0.974i)11-s + 12-s + (−0.222 + 0.974i)13-s + (−0.900 + 0.433i)14-s + (−0.900 + 0.433i)15-s + (−0.222 + 0.974i)16-s + 17-s + ⋯ |
L(s) = 1 | + (−0.900 − 0.433i)2-s + (0.623 − 0.781i)3-s + (0.623 + 0.781i)4-s + (−0.900 − 0.433i)5-s + (−0.900 + 0.433i)6-s + (0.623 − 0.781i)7-s + (−0.222 − 0.974i)8-s + (−0.222 − 0.974i)9-s + (0.623 + 0.781i)10-s + (−0.222 + 0.974i)11-s + 12-s + (−0.222 + 0.974i)13-s + (−0.900 + 0.433i)14-s + (−0.900 + 0.433i)15-s + (−0.222 + 0.974i)16-s + 17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.226 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.226 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4416425005 - 0.3507984638i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4416425005 - 0.3507984638i\) |
\(L(1)\) |
\(\approx\) |
\(0.6528919956 - 0.3316007111i\) |
\(L(1)\) |
\(\approx\) |
\(0.6528919956 - 0.3316007111i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 \) |
good | 2 | \( 1 + (-0.900 - 0.433i)T \) |
| 3 | \( 1 + (0.623 - 0.781i)T \) |
| 5 | \( 1 + (-0.900 - 0.433i)T \) |
| 7 | \( 1 + (0.623 - 0.781i)T \) |
| 11 | \( 1 + (-0.222 + 0.974i)T \) |
| 13 | \( 1 + (-0.222 + 0.974i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + (0.623 + 0.781i)T \) |
| 23 | \( 1 + (-0.900 + 0.433i)T \) |
| 31 | \( 1 + (-0.900 - 0.433i)T \) |
| 37 | \( 1 + (-0.222 - 0.974i)T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + (-0.900 + 0.433i)T \) |
| 47 | \( 1 + (-0.222 + 0.974i)T \) |
| 53 | \( 1 + (-0.900 - 0.433i)T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + (0.623 - 0.781i)T \) |
| 67 | \( 1 + (-0.222 - 0.974i)T \) |
| 71 | \( 1 + (-0.222 + 0.974i)T \) |
| 73 | \( 1 + (-0.900 + 0.433i)T \) |
| 79 | \( 1 + (-0.222 - 0.974i)T \) |
| 83 | \( 1 + (0.623 + 0.781i)T \) |
| 89 | \( 1 + (-0.900 - 0.433i)T \) |
| 97 | \( 1 + (0.623 + 0.781i)T \) |
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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
−37.5323488185379158211807198535, −36.56188202429477111490678345390, −34.71787583097152859425920222862, −34.33401996939299147590523320097, −32.65432489155710833939638683109, −31.590305978208988471360313284115, −30.13141659567623261227008763392, −28.13496151293009831090829527658, −27.3436392338942464677126124544, −26.461211388344044000001809037043, −25.17395709814187193342625292711, −23.931607996088498172389149957563, −22.08842207739531041879063235733, −20.523860204505413932175369198663, −19.3478096511209205288240911622, −18.216148454145388910059337790068, −16.30709588189565136484897237965, −15.36017234408214295049683752158, −14.39213514394595945283508975255, −11.54483671365899722668504897032, −10.31874047389810484869640923239, −8.63067405279540414287251963179, −7.75778970490669906986479694745, −5.37988265927970739855307872873, −2.96022208089554177309983676683,
1.61214558537285470986398597683, 3.848044766361780880431196628190, 7.313529678744304157539183855230, 7.97260225557239053948840210802, 9.61514612098094907729874945200, 11.57291529803550573585487992672, 12.60332597562818511398220771882, 14.48346948435541565153666979902, 16.31266312757757809585291694314, 17.73667931611430426524965741514, 19.00574120008196987920529678340, 20.09336404722979854887021504269, 20.86265942832055770752567184658, 23.361976142751984272507738870114, 24.458339000920987713214654022760, 25.87411516241852202531642179144, 26.953794204936854909656695631642, 28.179654925459393307074156217980, 29.59550062923938404567594411412, 30.68910156125391744068091843285, 31.52394370920901009923278116535, 33.66021342719269686084723334747, 35.05130917875897870589104782770, 36.19397129286534619407451083881, 36.54453494300704712246551213559