L(s) = 1 | + (0.631 − 0.775i)2-s + (−0.850 − 0.526i)3-s + (−0.201 − 0.979i)4-s + (−0.945 + 0.326i)5-s + (−0.945 + 0.326i)6-s + (−0.478 + 0.878i)7-s + (−0.886 − 0.462i)8-s + (0.445 + 0.895i)9-s + (−0.343 + 0.938i)10-s + (0.786 − 0.617i)11-s + (−0.343 + 0.938i)12-s + (0.932 − 0.361i)13-s + (0.378 + 0.925i)14-s + (0.975 + 0.219i)15-s + (−0.918 + 0.395i)16-s + (0.0184 + 0.999i)17-s + ⋯ |
L(s) = 1 | + (0.631 − 0.775i)2-s + (−0.850 − 0.526i)3-s + (−0.201 − 0.979i)4-s + (−0.945 + 0.326i)5-s + (−0.945 + 0.326i)6-s + (−0.478 + 0.878i)7-s + (−0.886 − 0.462i)8-s + (0.445 + 0.895i)9-s + (−0.343 + 0.938i)10-s + (0.786 − 0.617i)11-s + (−0.343 + 0.938i)12-s + (0.932 − 0.361i)13-s + (0.378 + 0.925i)14-s + (0.975 + 0.219i)15-s + (−0.918 + 0.395i)16-s + (0.0184 + 0.999i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0351i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0351i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01346937759 - 0.7664377107i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01346937759 - 0.7664377107i\) |
\(L(1)\) |
\(\approx\) |
\(0.6614004702 - 0.5137094696i\) |
\(L(1)\) |
\(\approx\) |
\(0.6614004702 - 0.5137094696i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 1021 | \( 1 \) |
good | 2 | \( 1 + (0.631 - 0.775i)T \) |
| 3 | \( 1 + (-0.850 - 0.526i)T \) |
| 5 | \( 1 + (-0.945 + 0.326i)T \) |
| 7 | \( 1 + (-0.478 + 0.878i)T \) |
| 11 | \( 1 + (0.786 - 0.617i)T \) |
| 13 | \( 1 + (0.932 - 0.361i)T \) |
| 17 | \( 1 + (0.0184 + 0.999i)T \) |
| 19 | \( 1 + (0.445 - 0.895i)T \) |
| 23 | \( 1 + (-0.918 - 0.395i)T \) |
| 29 | \( 1 + (-0.478 - 0.878i)T \) |
| 31 | \( 1 + (-0.713 + 0.700i)T \) |
| 37 | \( 1 + (0.997 - 0.0738i)T \) |
| 41 | \( 1 + (0.445 - 0.895i)T \) |
| 43 | \( 1 + (-0.128 - 0.991i)T \) |
| 47 | \( 1 + (0.378 + 0.925i)T \) |
| 53 | \( 1 + (-0.659 + 0.751i)T \) |
| 59 | \( 1 + (0.237 - 0.971i)T \) |
| 61 | \( 1 + (0.378 + 0.925i)T \) |
| 67 | \( 1 + (-0.809 + 0.587i)T \) |
| 71 | \( 1 + (-0.886 - 0.462i)T \) |
| 73 | \( 1 + (0.378 - 0.925i)T \) |
| 79 | \( 1 + (-0.999 - 0.0369i)T \) |
| 83 | \( 1 + (-0.850 - 0.526i)T \) |
| 89 | \( 1 + (0.739 - 0.673i)T \) |
| 97 | \( 1 + (-0.982 - 0.183i)T \) |
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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
−22.43579300350919116073224659646, −21.408573288089918113401824261987, −20.35553073162888213957452290432, −20.14913681438642644798994094470, −18.59222995367022699565394431505, −17.89772793653478419910933665849, −16.86376834623355078529350651134, −16.23207583960129996821810251387, −16.13057290875076602990580645764, −14.9950397711861442299014690141, −14.30172916434143397050007542275, −13.23562259381127278337115581923, −12.48442322461506031092537061288, −11.64078879106679966230964369089, −11.20586698987739653151875038065, −9.81211328038390008053621761477, −9.15306786274010652741299047491, −7.90323346712959292020335817088, −7.16062420434736672452015072170, −6.44448389087945163228209324871, −5.533359723623799151299397028309, −4.475408082735972317770920951583, −3.96810860857957892076478758402, −3.366002159923463260464523790864, −1.17609626989501702176222232892,
0.344534047888937092426867048437, 1.52527243197456638089857737942, 2.700548694331153748746929781223, 3.66477542693078986507590171536, 4.42366444086755162376143242969, 5.80791851073994290090061560612, 6.03566945258154398989688206800, 7.05144051136094518035224895002, 8.29714868838785446540177422725, 9.14775706417187450981563537334, 10.41253706128101212364170198557, 11.09394243291476435452353674659, 11.67687591577579417668358069614, 12.34043128351205607684855940169, 12.99237216538091151507047621548, 13.88045914681334601167021699077, 14.87947210525198000048364839062, 15.73363982839414071049476444984, 16.22407061891751313476779031896, 17.55655850105173813652647502706, 18.390242927624811212587353474064, 19.00163915023716016021524595841, 19.48379490908318166669792865358, 20.31417700991033677292574192810, 21.53917923102150926062523243380