L(s) = 1 | + (0.964 − 0.263i)2-s + (0.913 − 0.406i)3-s + (0.861 − 0.508i)4-s + (0.774 − 0.633i)6-s + (0.696 − 0.717i)8-s + (0.669 − 0.743i)9-s + (0.580 − 0.814i)12-s + (0.736 − 0.676i)13-s + (0.483 − 0.875i)16-s + (0.851 + 0.524i)17-s + (0.449 − 0.893i)18-s + (0.380 − 0.924i)19-s + (−0.981 − 0.189i)23-s + (0.345 − 0.938i)24-s + (0.532 − 0.846i)26-s + (0.309 − 0.951i)27-s + ⋯ |
L(s) = 1 | + (0.964 − 0.263i)2-s + (0.913 − 0.406i)3-s + (0.861 − 0.508i)4-s + (0.774 − 0.633i)6-s + (0.696 − 0.717i)8-s + (0.669 − 0.743i)9-s + (0.580 − 0.814i)12-s + (0.736 − 0.676i)13-s + (0.483 − 0.875i)16-s + (0.851 + 0.524i)17-s + (0.449 − 0.893i)18-s + (0.380 − 0.924i)19-s + (−0.981 − 0.189i)23-s + (0.345 − 0.938i)24-s + (0.532 − 0.846i)26-s + (0.309 − 0.951i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0971 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0971 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.658226278 - 4.032785926i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.658226278 - 4.032785926i\) |
\(L(1)\) |
\(\approx\) |
\(2.476977539 - 1.243598066i\) |
\(L(1)\) |
\(\approx\) |
\(2.476977539 - 1.243598066i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.964 - 0.263i)T \) |
| 3 | \( 1 + (0.913 - 0.406i)T \) |
| 13 | \( 1 + (0.736 - 0.676i)T \) |
| 17 | \( 1 + (0.851 + 0.524i)T \) |
| 19 | \( 1 + (0.380 - 0.924i)T \) |
| 23 | \( 1 + (-0.981 - 0.189i)T \) |
| 29 | \( 1 + (-0.941 + 0.336i)T \) |
| 31 | \( 1 + (0.595 - 0.803i)T \) |
| 37 | \( 1 + (-0.749 + 0.662i)T \) |
| 41 | \( 1 + (-0.362 - 0.931i)T \) |
| 43 | \( 1 + (-0.142 + 0.989i)T \) |
| 47 | \( 1 + (0.449 + 0.893i)T \) |
| 53 | \( 1 + (-0.483 - 0.875i)T \) |
| 59 | \( 1 + (0.625 + 0.780i)T \) |
| 61 | \( 1 + (-0.710 + 0.703i)T \) |
| 67 | \( 1 + (0.888 - 0.458i)T \) |
| 71 | \( 1 + (-0.564 + 0.825i)T \) |
| 73 | \( 1 + (-0.820 - 0.572i)T \) |
| 79 | \( 1 + (0.830 + 0.556i)T \) |
| 83 | \( 1 + (0.921 - 0.389i)T \) |
| 89 | \( 1 + (-0.723 - 0.690i)T \) |
| 97 | \( 1 + (-0.985 + 0.170i)T \) |
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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
−18.762435251947919235137747803, −17.89653472755344018985884415573, −16.72532857200429814119685503217, −16.383902354243870478518560062296, −15.70052780299770864367523397421, −15.15102952779720248979661028553, −14.23100499789636100719669641351, −14.00072388317155235955734696763, −13.433985280227602774633276146160, −12.47176384823410424894649998475, −11.93786168614017263536771705935, −11.109605856569219681775963964412, −10.2971320983332097134778004517, −9.64968709173745130817095184805, −8.72084024960518109328251465588, −8.046253820665402938677353158846, −7.44944180914598282129830647520, −6.672607733988028962889399266, −5.75599986351987093374903333361, −5.131598292549996095791107091027, −4.193994402766878263808939279761, −3.64900388778856770674870637017, −3.10088337039728229106852728851, −2.053853140439136472914769522833, −1.490188053207430903873390593433,
0.89442637538487742567647058470, 1.64906634777758995703005643994, 2.49219324701959125949395862693, 3.24620966188329793028480816747, 3.75995352118147407341821538939, 4.580349621710014791991368098898, 5.579210362190054369456859099726, 6.1772186548870474496777686679, 6.99097698268734970725270389532, 7.73379349388134651423690456525, 8.31229690525382042210141362383, 9.28967868039104325812712360327, 10.03529507189863681056538656028, 10.67310820148819956612245981499, 11.59216263355926097487443164216, 12.209365780905828280696593700495, 12.97437592069812110910567231816, 13.37968069888062013204592721307, 14.06364618768760232437212644798, 14.65968510094744623998267438840, 15.364606442189110882904179259733, 15.791187942022208715066290659457, 16.65631719463854467218709832495, 17.65888873607354334845229400910, 18.392217407038386995497361481140