L(s) = 1 | + (−0.342 + 0.939i)5-s + (0.342 + 0.939i)11-s + (0.984 − 0.173i)13-s + 17-s − i·19-s + (−0.173 − 0.984i)23-s + (−0.766 − 0.642i)25-s + (−0.984 − 0.173i)29-s + (0.766 − 0.642i)31-s + (0.866 + 0.5i)37-s + (−0.173 − 0.984i)41-s + (−0.642 + 0.766i)43-s + (−0.766 − 0.642i)47-s + (−0.866 − 0.5i)53-s − 55-s + ⋯ |
L(s) = 1 | + (−0.342 + 0.939i)5-s + (0.342 + 0.939i)11-s + (0.984 − 0.173i)13-s + 17-s − i·19-s + (−0.173 − 0.984i)23-s + (−0.766 − 0.642i)25-s + (−0.984 − 0.173i)29-s + (0.766 − 0.642i)31-s + (0.866 + 0.5i)37-s + (−0.173 − 0.984i)41-s + (−0.642 + 0.766i)43-s + (−0.766 − 0.642i)47-s + (−0.866 − 0.5i)53-s − 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.006833092410 + 0.01514381469i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.006833092410 + 0.01514381469i\) |
\(L(1)\) |
\(\approx\) |
\(0.9369636265 + 0.2083542226i\) |
\(L(1)\) |
\(\approx\) |
\(0.9369636265 + 0.2083542226i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.342 + 0.939i)T \) |
| 11 | \( 1 + (0.342 + 0.939i)T \) |
| 13 | \( 1 + (0.984 - 0.173i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - iT \) |
| 23 | \( 1 + (-0.173 - 0.984i)T \) |
| 29 | \( 1 + (-0.984 - 0.173i)T \) |
| 31 | \( 1 + (0.766 - 0.642i)T \) |
| 37 | \( 1 + (0.866 + 0.5i)T \) |
| 41 | \( 1 + (-0.173 - 0.984i)T \) |
| 43 | \( 1 + (-0.642 + 0.766i)T \) |
| 47 | \( 1 + (-0.766 - 0.642i)T \) |
| 53 | \( 1 + (-0.866 - 0.5i)T \) |
| 59 | \( 1 + (-0.984 + 0.173i)T \) |
| 61 | \( 1 + (0.642 - 0.766i)T \) |
| 67 | \( 1 + (-0.342 + 0.939i)T \) |
| 71 | \( 1 + (0.5 + 0.866i)T \) |
| 73 | \( 1 + (-0.5 - 0.866i)T \) |
| 79 | \( 1 + (0.939 - 0.342i)T \) |
| 83 | \( 1 + (-0.984 - 0.173i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (-0.766 - 0.642i)T \) |
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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
−19.21561532446567525823877579484, −18.50679325304591860037455603855, −17.68593686535595845790711496882, −16.86295044811338914081258462328, −16.38978840670682790764827950941, −15.76204452381900012168053187577, −15.071835766708965583549308778608, −14.041491428591249395073560132309, −13.499609925683843288525822648438, −12.87032050794561716981160209831, −11.99959984502640158359007514365, −11.37821632913475702889574865343, −10.836869830359524816459311178416, −9.59302555005258533152865918259, −9.17901338909712077130696852697, −8.30474585985372193481798047186, −7.864824353516494084527613296259, −6.81114477308474813745636983421, −5.922018854709914261573356084833, −5.33535802434748972518163400152, −4.43040329523358872755810906327, −3.62609940591382688461454547037, −2.99304233766077213546766626227, −1.51925086737132611925442864744, −1.04926227254675265547693430384,
0.00278367813663501076376971806, 1.26631917670373527495593294234, 2.125180783651738078057258462003, 3.11960801913886901575277476565, 3.79110739743970373698999864941, 4.49796133655463449833581484440, 5.66931866713769742031772473251, 6.3203947924008722393014501297, 7.007101955364407455237620659605, 7.90697601660303521737941169345, 8.301794408683782488134066996636, 9.599223361652210190165952332367, 10.03558043930233735336843069686, 10.80361542425302639231720535588, 11.55558070192253348324495349861, 12.17516013134203320967153911092, 12.951134706030861648286327836361, 13.81928746115333578462857789145, 14.64071379531563332888856415278, 14.906854705483733815613053143712, 15.79225124103383710523495164005, 16.51368643132243216952697419192, 17.233375234316548114806170075343, 18.135001900778899125373765580403, 18.623187079932611456415039691538