L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s − i·8-s + (0.5 + 0.866i)11-s + i·13-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + (0.5 − 0.866i)19-s − i·22-s + (0.866 + 0.5i)23-s + (0.5 − 0.866i)26-s + 29-s + (−0.5 − 0.866i)31-s + (0.866 − 0.5i)32-s − 34-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s − i·8-s + (0.5 + 0.866i)11-s + i·13-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + (0.5 − 0.866i)19-s − i·22-s + (0.866 + 0.5i)23-s + (0.5 − 0.866i)26-s + 29-s + (−0.5 − 0.866i)31-s + (0.866 − 0.5i)32-s − 34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7245183557 - 0.03384175778i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7245183557 - 0.03384175778i\) |
\(L(1)\) |
\(\approx\) |
\(0.7576548819 - 0.07122149674i\) |
\(L(1)\) |
\(\approx\) |
\(0.7576548819 - 0.07122149674i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 + iT \) |
| 17 | \( 1 + (0.866 - 0.5i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.866 + 0.5i)T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (0.866 + 0.5i)T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + (-0.866 - 0.5i)T \) |
| 53 | \( 1 + (-0.866 + 0.5i)T \) |
| 59 | \( 1 + (-0.5 - 0.866i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.866 + 0.5i)T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + (0.866 - 0.5i)T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 - iT \) |
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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
−29.4858983106391343056817902926, −28.60820850899327563625922261152, −27.32312555731768786650359891353, −26.94289898659468712667088893775, −25.47611825911802291192644348153, −24.89826443727250078963050328478, −23.743094486931191291655557800681, −22.70928856574814403799326967201, −21.242299842092793180847808790229, −20.0672947012960620364208660701, −19.097675552083183836941560181775, −18.14959623548031518225176802636, −16.998568297245835424286771940567, −16.18665565203927395756271287769, −14.960093579887217963849415224198, −14.00616695998076775804494791123, −12.34831430505831956525643310329, −10.96892090701662387114494545513, −10.01339171595318774215682726944, −8.69738344833147717050250361244, −7.77737889384323535001617498221, −6.373718433653193559247897545431, −5.308214054393170365763837959512, −3.19293580887404261068090611130, −1.19758382813300735391753483741,
1.47323464074645274625666854748, 3.016202078233119048861531401704, 4.586250112438853644447667121938, 6.64717638898275485498567875126, 7.637885440196233877385623629344, 9.12461780846155767598867902407, 9.83549612096686191013846472978, 11.312534087556372437868991241311, 12.08685444665869354861064790575, 13.422704495135924466958262428993, 14.92283190475603011232828219970, 16.23780958441153720636823281645, 17.16717813129192940903904541783, 18.20193458217397866778104516995, 19.22618639482586948998590025545, 20.16981696410980663618312019745, 21.18539624639958458011387346696, 22.1846197347678109666745806876, 23.513334180126688196391328656959, 24.91361033941547509528063330360, 25.721557887483261503833025603161, 26.7550832137935384163871315993, 27.71862002703768501841729225657, 28.59037487945380627964209707515, 29.52307112175659053127358579200