L(s) = 1 | + (0.866 − 0.5i)3-s + (−0.5 + 0.866i)7-s + (0.5 − 0.866i)9-s + (−0.866 + 0.5i)11-s + (0.866 + 0.5i)17-s + (0.866 + 0.5i)19-s + i·21-s + (−0.866 + 0.5i)23-s − i·27-s + (0.5 + 0.866i)29-s − i·31-s + (−0.5 + 0.866i)33-s + (0.5 + 0.866i)37-s + (−0.866 + 0.5i)41-s + (0.866 + 0.5i)43-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)3-s + (−0.5 + 0.866i)7-s + (0.5 − 0.866i)9-s + (−0.866 + 0.5i)11-s + (0.866 + 0.5i)17-s + (0.866 + 0.5i)19-s + i·21-s + (−0.866 + 0.5i)23-s − i·27-s + (0.5 + 0.866i)29-s − i·31-s + (−0.5 + 0.866i)33-s + (0.5 + 0.866i)37-s + (−0.866 + 0.5i)41-s + (0.866 + 0.5i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.822758708 + 1.041899411i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.822758708 + 1.041899411i\) |
\(L(1)\) |
\(\approx\) |
\(1.312810804 + 0.1312926369i\) |
\(L(1)\) |
\(\approx\) |
\(1.312810804 + 0.1312926369i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.866 + 0.5i)T \) |
| 17 | \( 1 + (0.866 + 0.5i)T \) |
| 19 | \( 1 + (0.866 + 0.5i)T \) |
| 23 | \( 1 + (-0.866 + 0.5i)T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 - iT \) |
| 37 | \( 1 + (0.5 + 0.866i)T \) |
| 41 | \( 1 + (-0.866 + 0.5i)T \) |
| 43 | \( 1 + (0.866 + 0.5i)T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 - iT \) |
| 59 | \( 1 + (-0.866 - 0.5i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.866 - 0.5i)T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (-0.866 + 0.5i)T \) |
| 97 | \( 1 + (-0.5 + 0.866i)T \) |
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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
−25.75854807289780962058229277133, −24.68783112653183236055526368392, −23.74692921812292220147180102911, −22.7074999645223900067723451944, −21.75524911904627252097887331839, −20.72746334619556489974935915217, −20.18806815173830763044901536728, −19.16698664354184127718295338909, −18.3456502801372696445070517136, −16.901814151686133411270926231353, −16.08334618624184817863400168235, −15.36634406328652819781028537285, −13.90547121117239172820511183602, −13.72479566428842979748464616618, −12.42370290163529432160450438350, −10.94723610940759706655452771435, −10.09773417040597860770460700691, −9.31318671453022153201598954318, −7.99407289381297991648867681743, −7.35156922749827283992258296057, −5.77821712462214185087862184479, −4.483167533990135843535419873102, −3.44140664429480418492133401363, −2.46719053514251435929966550265, −0.60087705055952565328322565218,
1.42268606674504629986636616148, 2.65682201853045912544269988971, 3.53886124626226544077745287818, 5.20230907074784218563422733588, 6.341908378475445631591330873497, 7.56515215377453961376923581436, 8.34278082406918272749321932905, 9.47159333224783574392736922190, 10.25274582633549496868766473834, 12.013548941639456545658188983372, 12.553846007747722091441037467961, 13.58473234802728342005203243606, 14.56793760519405529815636741828, 15.464248701606341753341851827003, 16.27540252963815161535014430299, 17.88506912559648435351076964497, 18.457631948575509011702718271188, 19.34774583802021232037598951193, 20.23167592092980421483689462168, 21.14421100312114675589791487467, 22.04017089853320294222504821668, 23.290965747288920591652626400624, 24.02266122005873726904885112613, 25.206168569763524261480117580739, 25.583928323700287961963532751816