L(s) = 1 | + (−0.984 + 0.173i)2-s + (−0.0581 − 0.998i)3-s + (0.939 − 0.342i)4-s + (−0.597 + 0.802i)5-s + (0.230 + 0.973i)6-s + (0.396 + 0.918i)7-s + (−0.866 + 0.5i)8-s + (−0.993 + 0.116i)9-s + (0.448 − 0.893i)10-s + (0.448 + 0.893i)11-s + (−0.396 − 0.918i)12-s + (0.918 − 0.396i)13-s + (−0.549 − 0.835i)14-s + (0.835 + 0.549i)15-s + (0.766 − 0.642i)16-s + (0.642 + 0.766i)17-s + ⋯ |
L(s) = 1 | + (−0.984 + 0.173i)2-s + (−0.0581 − 0.998i)3-s + (0.939 − 0.342i)4-s + (−0.597 + 0.802i)5-s + (0.230 + 0.973i)6-s + (0.396 + 0.918i)7-s + (−0.866 + 0.5i)8-s + (−0.993 + 0.116i)9-s + (0.448 − 0.893i)10-s + (0.448 + 0.893i)11-s + (−0.396 − 0.918i)12-s + (0.918 − 0.396i)13-s + (−0.549 − 0.835i)14-s + (0.835 + 0.549i)15-s + (0.766 − 0.642i)16-s + (0.642 + 0.766i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.623 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.623 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3699113552 + 0.7681907548i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3699113552 + 0.7681907548i\) |
\(L(1)\) |
\(\approx\) |
\(0.6562860785 + 0.07880881983i\) |
\(L(1)\) |
\(\approx\) |
\(0.6562860785 + 0.07880881983i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (-0.984 + 0.173i)T \) |
| 3 | \( 1 + (-0.0581 - 0.998i)T \) |
| 5 | \( 1 + (-0.597 + 0.802i)T \) |
| 7 | \( 1 + (0.396 + 0.918i)T \) |
| 11 | \( 1 + (0.448 + 0.893i)T \) |
| 13 | \( 1 + (0.918 - 0.396i)T \) |
| 17 | \( 1 + (0.642 + 0.766i)T \) |
| 19 | \( 1 + (-0.642 - 0.766i)T \) |
| 23 | \( 1 + (0.984 + 0.173i)T \) |
| 29 | \( 1 + (-0.0581 - 0.998i)T \) |
| 31 | \( 1 + (-0.993 + 0.116i)T \) |
| 41 | \( 1 - iT \) |
| 43 | \( 1 + (0.766 - 0.642i)T \) |
| 47 | \( 1 + (0.727 + 0.686i)T \) |
| 53 | \( 1 + (-0.116 + 0.993i)T \) |
| 59 | \( 1 + (-0.549 + 0.835i)T \) |
| 61 | \( 1 + (-0.286 + 0.957i)T \) |
| 67 | \( 1 + (-0.802 + 0.597i)T \) |
| 71 | \( 1 + (-0.173 + 0.984i)T \) |
| 73 | \( 1 + (0.893 - 0.448i)T \) |
| 79 | \( 1 + (-0.918 - 0.396i)T \) |
| 83 | \( 1 + (0.0581 + 0.998i)T \) |
| 89 | \( 1 + (0.286 - 0.957i)T \) |
| 97 | \( 1 + (-0.597 + 0.802i)T \) |
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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
−18.02555063974860422876838117297, −17.01905640128718024603046998331, −16.61361226828068390623982453739, −16.39382127522753252098912131365, −15.68388147489828560654731388283, −14.75595868051151241767677849104, −14.19943019593867133321917532389, −13.200938845798396794193025983513, −12.31151626736654473540556600212, −11.49116635763332894241129558539, −11.0268622925514957445970138714, −10.62761195011271046121933449813, −9.557028936708225985115443214690, −9.096800296120928508647820341222, −8.404162728059059732338335083410, −7.9232773224592799227152771717, −6.98464930616030284179318454983, −6.09881935708974535248381103907, −5.23002664142960327164212344873, −4.35379508661250275552272247792, −3.570918631232295707061970533, −3.22233311388377462748225490966, −1.67138329669606952236476475407, −0.9397044798594293281291777238, −0.23949910042515542742138429398,
0.87791468130987308921405591715, 1.69020067283907004879160833897, 2.433564339847226711277417539296, 3.06722356303258947306895371408, 4.194820594767122843214412026482, 5.62250889349628234240929692933, 5.985153898491631011709039444662, 6.88030970043368168545202952388, 7.37891919923789360559516132735, 8.009892688060030528488019635189, 8.75842360700633133408954409109, 9.20886743444698557545414642203, 10.48845235116413204805814426723, 10.93310351973156463095308872125, 11.608901207501770287002258899189, 12.23649963133437140051924169914, 12.78428062885683719313039872527, 13.930785102933442430250595572614, 14.75714012667811680732833405008, 15.174436124504586295128594271719, 15.66453817675059081476190175606, 16.79401323659780862846016572997, 17.47766295756853941683698785820, 17.88003661033285508109839484972, 18.60095194350175993612316047718