L(s) = 1 | + (−0.724 + 0.689i)2-s + (−0.442 + 0.896i)3-s + (0.0487 − 0.998i)4-s + (−0.696 + 0.717i)5-s + (−0.297 − 0.954i)6-s + (0.981 − 0.193i)7-s + (0.653 + 0.756i)8-s + (−0.608 − 0.793i)9-s + (0.00975 − 0.999i)10-s + (0.909 − 0.416i)11-s + (0.874 + 0.485i)12-s + (0.909 + 0.416i)13-s + (−0.576 + 0.816i)14-s + (−0.334 − 0.942i)15-s + (−0.995 − 0.0974i)16-s + (−0.844 − 0.536i)17-s + ⋯ |
L(s) = 1 | + (−0.724 + 0.689i)2-s + (−0.442 + 0.896i)3-s + (0.0487 − 0.998i)4-s + (−0.696 + 0.717i)5-s + (−0.297 − 0.954i)6-s + (0.981 − 0.193i)7-s + (0.653 + 0.756i)8-s + (−0.608 − 0.793i)9-s + (0.00975 − 0.999i)10-s + (0.909 − 0.416i)11-s + (0.874 + 0.485i)12-s + (0.909 + 0.416i)13-s + (−0.576 + 0.816i)14-s + (−0.334 − 0.942i)15-s + (−0.995 − 0.0974i)16-s + (−0.844 − 0.536i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.182 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.182 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5389115724 + 0.6484729315i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5389115724 + 0.6484729315i\) |
\(L(1)\) |
\(\approx\) |
\(0.5700401868 + 0.3857576279i\) |
\(L(1)\) |
\(\approx\) |
\(0.5700401868 + 0.3857576279i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 967 | \( 1 \) |
good | 2 | \( 1 + (-0.724 + 0.689i)T \) |
| 3 | \( 1 + (-0.442 + 0.896i)T \) |
| 5 | \( 1 + (-0.696 + 0.717i)T \) |
| 7 | \( 1 + (0.981 - 0.193i)T \) |
| 11 | \( 1 + (0.909 - 0.416i)T \) |
| 13 | \( 1 + (0.909 + 0.416i)T \) |
| 17 | \( 1 + (-0.844 - 0.536i)T \) |
| 19 | \( 1 + (-0.977 + 0.212i)T \) |
| 23 | \( 1 + (-0.477 + 0.878i)T \) |
| 29 | \( 1 + (-0.999 + 0.0195i)T \) |
| 31 | \( 1 + (0.592 - 0.805i)T \) |
| 37 | \( 1 + (0.892 - 0.451i)T \) |
| 41 | \( 1 + (0.682 + 0.730i)T \) |
| 43 | \( 1 + (0.993 - 0.116i)T \) |
| 47 | \( 1 + (0.527 - 0.849i)T \) |
| 53 | \( 1 + (0.962 + 0.269i)T \) |
| 59 | \( 1 + (0.389 - 0.921i)T \) |
| 61 | \( 1 + (0.682 + 0.730i)T \) |
| 67 | \( 1 + (0.592 + 0.805i)T \) |
| 71 | \( 1 + (-0.724 - 0.689i)T \) |
| 73 | \( 1 + (0.962 - 0.269i)T \) |
| 79 | \( 1 + (-0.750 + 0.660i)T \) |
| 83 | \( 1 + (-0.864 + 0.502i)T \) |
| 89 | \( 1 + (0.592 + 0.805i)T \) |
| 97 | \( 1 + (-0.900 - 0.433i)T \) |
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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
−21.32993028299815148737929083981, −20.45398775332771389011942907531, −19.92270514049243470949895036420, −19.16612573523771826096534245718, −18.43178963213561740290256849028, −17.55322909472850915106094864981, −17.22039711262226734726968252310, −16.313120323647229306631869107184, −15.35437984515346471762417879381, −14.227938973013058663535020231054, −13.05904523073353557602860752861, −12.60285635049461667487698290612, −11.79256437199738098555560034178, −11.18938346398582080671509503040, −10.57563198203512062023921337968, −8.98284480887601351666224029548, −8.52777610908246187621507385630, −7.86857501610220582584710752658, −6.9527116177293030003660849268, −5.94489344893815726828107658713, −4.55845869082532587713468028133, −3.99408943777925357639783585220, −2.40569730603480087806219032828, −1.57620716452512136597560971900, −0.75968640661902128126806297683,
0.833678810730381836629148157228, 2.253881768548925899806591820898, 3.98340855347599231378508756998, 4.255496170917498320393255899834, 5.623644810883162115095946620728, 6.32730480904055236945428120852, 7.199903619528222446603909612789, 8.21673083247852963974201664787, 8.8922181754415035225860546154, 9.75504960120433762842585031858, 10.88027278719576263672122736856, 11.204451183791702333106972640232, 11.740507278083219251192239661097, 13.622289855694560296260748116116, 14.45681121039774367300225296164, 14.98330553986465987939054704489, 15.70567085891415571614876056090, 16.43906584908103498744269611873, 17.18438264589268456953603340738, 17.919644458244486888073560240181, 18.617496394612449685201127458353, 19.55527968576430757512890189085, 20.28120173377377734879740089559, 21.17732283771176895453446252804, 22.100593290995442207816404618223