L(s) = 1 | + (0.965 + 0.258i)2-s + (0.866 + 0.5i)4-s + (0.608 + 0.793i)5-s + (−0.608 + 0.793i)7-s + (0.707 + 0.707i)8-s + (0.382 + 0.923i)10-s + (−0.793 − 0.608i)11-s + (−0.866 − 0.5i)13-s + (−0.793 + 0.608i)14-s + (0.5 + 0.866i)16-s + (0.707 − 0.707i)19-s + (0.130 + 0.991i)20-s + (−0.608 − 0.793i)22-s + (0.130 − 0.991i)23-s + (−0.258 + 0.965i)25-s + (−0.707 − 0.707i)26-s + ⋯ |
L(s) = 1 | + (0.965 + 0.258i)2-s + (0.866 + 0.5i)4-s + (0.608 + 0.793i)5-s + (−0.608 + 0.793i)7-s + (0.707 + 0.707i)8-s + (0.382 + 0.923i)10-s + (−0.793 − 0.608i)11-s + (−0.866 − 0.5i)13-s + (−0.793 + 0.608i)14-s + (0.5 + 0.866i)16-s + (0.707 − 0.707i)19-s + (0.130 + 0.991i)20-s + (−0.608 − 0.793i)22-s + (0.130 − 0.991i)23-s + (−0.258 + 0.965i)25-s + (−0.707 − 0.707i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.676567599 + 0.9810933415i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.676567599 + 0.9810933415i\) |
\(L(1)\) |
\(\approx\) |
\(1.658780815 + 0.5958656424i\) |
\(L(1)\) |
\(\approx\) |
\(1.658780815 + 0.5958656424i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
good | 2 | \( 1 + (0.965 + 0.258i)T \) |
| 5 | \( 1 + (0.608 + 0.793i)T \) |
| 7 | \( 1 + (-0.608 + 0.793i)T \) |
| 11 | \( 1 + (-0.793 - 0.608i)T \) |
| 13 | \( 1 + (-0.866 - 0.5i)T \) |
| 19 | \( 1 + (0.707 - 0.707i)T \) |
| 23 | \( 1 + (0.130 - 0.991i)T \) |
| 29 | \( 1 + (0.991 - 0.130i)T \) |
| 31 | \( 1 + (0.793 - 0.608i)T \) |
| 37 | \( 1 + (0.923 + 0.382i)T \) |
| 41 | \( 1 + (-0.991 - 0.130i)T \) |
| 43 | \( 1 + (0.258 - 0.965i)T \) |
| 47 | \( 1 + (-0.866 + 0.5i)T \) |
| 53 | \( 1 + (-0.707 + 0.707i)T \) |
| 59 | \( 1 + (-0.965 + 0.258i)T \) |
| 61 | \( 1 + (0.608 - 0.793i)T \) |
| 67 | \( 1 + (0.5 - 0.866i)T \) |
| 71 | \( 1 + (-0.923 - 0.382i)T \) |
| 73 | \( 1 + (-0.382 + 0.923i)T \) |
| 79 | \( 1 + (0.793 + 0.608i)T \) |
| 83 | \( 1 + (-0.965 - 0.258i)T \) |
| 89 | \( 1 + iT \) |
| 97 | \( 1 + (-0.991 + 0.130i)T \) |
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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
−28.30308168251853489014472649959, −26.79642368020036863910742066027, −25.59085913570000059822141619562, −24.810148916904659348346157840096, −23.72070180087435979560273584676, −23.05697726172469753745119682629, −21.86541061973260119260182295768, −21.01088472671678041109593034682, −20.152666610628292915864068217212, −19.361954128662639411535320523124, −17.73722394051489776356110929950, −16.58106051855670202643713947496, −15.833627455874564804368463735052, −14.43039819630550434527108533303, −13.51138972462191915943001367639, −12.759321578363276138133217892087, −11.80141800266214178836088665829, −10.2141556863124422880977983152, −9.67871150056388614243000556730, −7.71735195223579691138434120823, −6.571920651616378189620483475874, −5.27834287991799183077867948103, −4.4085730391532669451162189312, −2.92404501065846236497802404709, −1.477784050020156294946687180521,
2.55058068640847325757443978773, 3.01449290559805806018959784671, 4.93737088112240791928087406358, 5.92566368226910697250216663830, 6.83243203522950272550299561417, 8.1354016644782720287163028929, 9.776457642537139118196014148253, 10.86360576108894904297991443631, 12.07008140494648593078843822340, 13.10609472626361702102891306965, 13.974743408171808004109491393936, 15.092396555924141185437230875096, 15.788484800163754478186342270769, 17.0415326944233934127512142048, 18.23382856348918669602862835007, 19.27739430142471264111851896090, 20.56658430856056746471043623275, 21.74390396675058731515090981374, 22.14019964767608877705835312745, 23.083250175626917110935628856232, 24.35441228616177638261690740302, 25.10266392587556046201388123369, 26.03900820291663544018650035166, 26.81610472192529070234947241789, 28.70179211611950792398548147794