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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.811389767 - 2.120105693i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.811389767 - 2.120105693i\) |
\(L(1)\) |
\(\approx\) |
\(1.558988302 - 0.8671390436i\) |
\(L(1)\) |
\(\approx\) |
\(1.558988302 - 0.8671390436i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
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
−29.99254568077376053088559787444, −28.93765441335301439070310883678, −27.62914842289857164076855848802, −26.37627833044666502072135912074, −25.43372122304953932068851038632, −24.587040105164494436134299761983, −23.397069710082722830759158485108, −22.71825707983812011733009545939, −21.50320556327192892116736238898, −20.68541964978492134831108076776, −19.42611725196674377781380428766, −17.93747792327814077191974012194, −16.784826766044178160207265917338, −15.922607923618600043139464964, −14.579262474619503434295432482655, −13.98144223835120123844049983909, −12.44002784875433066084308874638, −11.82649685111012507425664641583, −10.16653743773725181359665410326, −8.613216779714191651174558326139, −7.27801141727382561322340482392, −6.26738410863518089795006908046, −4.83431089499032463264806659654, −3.73164202577169353113149144993, −2.034247183860731884092781772548,
0.97152758335965067345692074834, 2.763489543264713663393180448041, 3.962061405869019912897533012534, 5.43547563572545142456145900971, 6.45733475462387093109276529267, 8.13770839572759376426490657176, 9.77315740806023969448895588, 10.869241114657138087503112804798, 11.940327208715621195576407937241, 13.04730583625022113026095806874, 14.05444977457702051226449324128, 15.119154594873304106534882632861, 16.18914085650522333433797709483, 17.61784432691752959817640987837, 19.08716537654744831637364527077, 19.80112472080749755482284656710, 21.03774849337250959819659031050, 21.87356702915152994372663677151, 22.85279667960204528940062901071, 23.901151945483155487532745908009, 24.77700019899877822262600810081, 25.938285959553782568027316106607, 27.51279374403017242299167233018, 28.1711248653568268107540459891, 29.65162769126381317801274013190