L(s) = 1 | + (0.917 − 0.398i)2-s + (0.406 + 0.913i)3-s + (0.683 − 0.730i)4-s + (0.736 + 0.676i)6-s + (0.336 − 0.941i)8-s + (−0.669 + 0.743i)9-s + (0.945 + 0.327i)12-s + (0.856 + 0.516i)13-s + (−0.0665 − 0.997i)16-s + (0.986 − 0.161i)17-s + (−0.318 + 0.948i)18-s + (0.449 + 0.893i)19-s + (−0.371 − 0.928i)23-s + (0.997 − 0.0760i)24-s + (0.991 + 0.132i)26-s + (−0.951 − 0.309i)27-s + ⋯ |
L(s) = 1 | + (0.917 − 0.398i)2-s + (0.406 + 0.913i)3-s + (0.683 − 0.730i)4-s + (0.736 + 0.676i)6-s + (0.336 − 0.941i)8-s + (−0.669 + 0.743i)9-s + (0.945 + 0.327i)12-s + (0.856 + 0.516i)13-s + (−0.0665 − 0.997i)16-s + (0.986 − 0.161i)17-s + (−0.318 + 0.948i)18-s + (0.449 + 0.893i)19-s + (−0.371 − 0.928i)23-s + (0.997 − 0.0760i)24-s + (0.991 + 0.132i)26-s + (−0.951 − 0.309i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(4.177304983 + 0.3518341129i\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.177304983 + 0.3518341129i\) |
\(L(1)\) |
\(\approx\) |
\(2.230514659 + 0.05686310305i\) |
\(L(1)\) |
\(\approx\) |
\(2.230514659 + 0.05686310305i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.917 - 0.398i)T \) |
| 3 | \( 1 + (0.406 + 0.913i)T \) |
| 13 | \( 1 + (0.856 + 0.516i)T \) |
| 17 | \( 1 + (0.986 - 0.161i)T \) |
| 19 | \( 1 + (0.449 + 0.893i)T \) |
| 23 | \( 1 + (-0.371 - 0.928i)T \) |
| 29 | \( 1 + (0.362 - 0.931i)T \) |
| 31 | \( 1 + (-0.797 + 0.603i)T \) |
| 37 | \( 1 + (-0.424 - 0.905i)T \) |
| 41 | \( 1 + (0.870 - 0.491i)T \) |
| 43 | \( 1 + (0.281 + 0.959i)T \) |
| 47 | \( 1 + (0.318 + 0.948i)T \) |
| 53 | \( 1 + (0.997 + 0.0665i)T \) |
| 59 | \( 1 + (0.861 - 0.508i)T \) |
| 61 | \( 1 + (0.595 + 0.803i)T \) |
| 67 | \( 1 + (-0.814 + 0.580i)T \) |
| 71 | \( 1 + (-0.254 - 0.967i)T \) |
| 73 | \( 1 + (-0.0380 + 0.999i)T \) |
| 79 | \( 1 + (0.851 - 0.524i)T \) |
| 83 | \( 1 + (0.170 - 0.985i)T \) |
| 89 | \( 1 + (0.0475 - 0.998i)T \) |
| 97 | \( 1 + (-0.825 + 0.564i)T \) |
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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
−18.10582513659535538642214010016, −17.85008741504672200911218719210, −16.88441384205669985505049813543, −16.29010017400572608410892022329, −15.34207330992353846879390229529, −15.01799672833870612576478719602, −14.002538763053631231065661501861, −13.73294918249967315250034521169, −12.99843015079472189064144351299, −12.45723249921643746926592709806, −11.71217471846756860652078175370, −11.17446827052337005968525600572, −10.204908337287413185436116440682, −9.10037336121192160109008275385, −8.431307836284317808464055953, −7.73136252311468241509357576931, −7.177092892679329695558451395319, −6.48548814322428115373767353457, −5.6257234994142829434525585602, −5.27482762576276987832276296820, −3.937419503161996540743832652409, −3.402830802273645836799162320647, −2.71063921637364556965141304135, −1.79074568740550196414712249331, −0.9362420254087235157133389555,
0.984344197452008404183316274055, 2.002679289877159479983742239, 2.765345181661030779639370542921, 3.6180840354828356898400408952, 4.03117441301678991020369500767, 4.79906387419605133478702960310, 5.71040379080622060615615833195, 6.05312745347354983085346683407, 7.22311380262548516732615946027, 7.98196789805278430612791240847, 8.89351087619764683257568831342, 9.60117274040743558792424690620, 10.33117925949624749018838296427, 10.77645294330873931681978480606, 11.62265777173774884693844156686, 12.18615075823751084217734183806, 13.042014229256014565518241081240, 13.81426082815020933001192213824, 14.48080293940944200166589962269, 14.604974605764513618986185045430, 15.78180132939892186643129608785, 16.15806624547484513939557391983, 16.56991395042595444776354612324, 17.74019333666297635994329741753, 18.69323543042317279449158016711