L(s) = 1 | + (−0.526 + 0.850i)2-s + (−0.739 + 0.673i)3-s + (−0.445 − 0.895i)4-s + (−0.982 + 0.183i)5-s + (−0.183 − 0.982i)6-s + (−0.961 − 0.273i)7-s + (0.995 + 0.0922i)8-s + (0.0922 − 0.995i)9-s + (0.361 − 0.932i)10-s + (0.932 − 0.361i)11-s + (0.932 + 0.361i)12-s + (0.526 + 0.850i)13-s + (0.739 − 0.673i)14-s + (0.602 − 0.798i)15-s + (−0.602 + 0.798i)16-s + (0.982 − 0.183i)17-s + ⋯ |
L(s) = 1 | + (−0.526 + 0.850i)2-s + (−0.739 + 0.673i)3-s + (−0.445 − 0.895i)4-s + (−0.982 + 0.183i)5-s + (−0.183 − 0.982i)6-s + (−0.961 − 0.273i)7-s + (0.995 + 0.0922i)8-s + (0.0922 − 0.995i)9-s + (0.361 − 0.932i)10-s + (0.932 − 0.361i)11-s + (0.932 + 0.361i)12-s + (0.526 + 0.850i)13-s + (0.739 − 0.673i)14-s + (0.602 − 0.798i)15-s + (−0.602 + 0.798i)16-s + (0.982 − 0.183i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6506306527 + 0.02282993296i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6506306527 + 0.02282993296i\) |
\(L(1)\) |
\(\approx\) |
\(0.4877084726 + 0.2229313467i\) |
\(L(1)\) |
\(\approx\) |
\(0.4877084726 + 0.2229313467i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 1021 | \( 1 \) |
good | 2 | \( 1 + (-0.526 + 0.850i)T \) |
| 3 | \( 1 + (-0.739 + 0.673i)T \) |
| 5 | \( 1 + (-0.982 + 0.183i)T \) |
| 7 | \( 1 + (-0.961 - 0.273i)T \) |
| 11 | \( 1 + (0.932 - 0.361i)T \) |
| 13 | \( 1 + (0.526 + 0.850i)T \) |
| 17 | \( 1 + (0.982 - 0.183i)T \) |
| 19 | \( 1 + (0.995 - 0.0922i)T \) |
| 23 | \( 1 + (-0.602 - 0.798i)T \) |
| 29 | \( 1 + (0.273 + 0.961i)T \) |
| 31 | \( 1 + (0.995 + 0.0922i)T \) |
| 37 | \( 1 + (-0.673 - 0.739i)T \) |
| 41 | \( 1 + (-0.0922 - 0.995i)T \) |
| 43 | \( 1 + (-0.961 - 0.273i)T \) |
| 47 | \( 1 + (0.739 - 0.673i)T \) |
| 53 | \( 1 + (-0.798 + 0.602i)T \) |
| 59 | \( 1 + (-0.673 - 0.739i)T \) |
| 61 | \( 1 + (-0.739 + 0.673i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (0.0922 - 0.995i)T \) |
| 73 | \( 1 + (0.739 + 0.673i)T \) |
| 79 | \( 1 + (-0.932 - 0.361i)T \) |
| 83 | \( 1 + (-0.739 + 0.673i)T \) |
| 89 | \( 1 + (0.445 - 0.895i)T \) |
| 97 | \( 1 + (0.961 + 0.273i)T \) |
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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
−21.45875530837850941460024307320, −20.233614542274663279355384272358, −19.80127403474596785206604416879, −19.004725878574483910057150079543, −18.57994281887335647686238668842, −17.58032634300473350833867070693, −16.92015592147088148882477155439, −16.1176252689477776093753267942, −15.462395234795735511335058138539, −13.93259141396144180371895135572, −13.14297295878345547060637319767, −12.290530037858886351219483186602, −11.94158490578287738455164632508, −11.28491647477802213072244042599, −10.15530230648959831927781668612, −9.57638721967743672538080112527, −8.28840717957950035523504424965, −7.78926674145129239579382882408, −6.84366764111969252705578025511, −5.86602489325065154495843763179, −4.7037781348386507642083679361, −3.6104172544311815792489600471, −2.957449474166002755974464407861, −1.439156046317961773218785505685, −0.76147350526838893813802651447,
0.32149653876621351735285012236, 1.14486390801539930996934504513, 3.36802295108183733435334525570, 3.967222112962715657622845452551, 4.927564859526204504830983467595, 6.00795112261626205040499840897, 6.68116273975620061662085528711, 7.30638511787378612087079873869, 8.57935410739037559246946981002, 9.23320777483819327228631329279, 10.099225348362749682479726151192, 10.79493884972611471230855303773, 11.76583139406545030909083118722, 12.377159570063596838419993086520, 13.93549124620435560463402552606, 14.37705195036047792769312531472, 15.60914527415794602965282009417, 15.93410935233151650165781435637, 16.62409714753654097452403736655, 17.08328291035500677627089245617, 18.38483094096860052987101539952, 18.80475616847969783661314712906, 19.73308775038782084205722466704, 20.37761790453874806645985624063, 21.69333658228033148188493471599