L(s) = 1 | + (0.695 − 0.718i)2-s + (−0.538 − 0.842i)3-s + (−0.0334 − 0.999i)4-s + (−0.420 − 0.907i)5-s + (−0.979 − 0.199i)6-s + (−0.645 + 0.763i)7-s + (−0.741 − 0.670i)8-s + (−0.420 + 0.907i)9-s + (−0.944 − 0.328i)10-s + (−0.0334 + 0.999i)11-s + (−0.824 + 0.565i)12-s + (−0.979 − 0.199i)13-s + (0.100 + 0.994i)14-s + (−0.538 + 0.842i)15-s + (−0.997 + 0.0667i)16-s + (0.100 − 0.994i)17-s + ⋯ |
L(s) = 1 | + (0.695 − 0.718i)2-s + (−0.538 − 0.842i)3-s + (−0.0334 − 0.999i)4-s + (−0.420 − 0.907i)5-s + (−0.979 − 0.199i)6-s + (−0.645 + 0.763i)7-s + (−0.741 − 0.670i)8-s + (−0.420 + 0.907i)9-s + (−0.944 − 0.328i)10-s + (−0.0334 + 0.999i)11-s + (−0.824 + 0.565i)12-s + (−0.979 − 0.199i)13-s + (0.100 + 0.994i)14-s + (−0.538 + 0.842i)15-s + (−0.997 + 0.0667i)16-s + (0.100 − 0.994i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.2749776402 - 0.4111477345i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.2749776402 - 0.4111477345i\) |
\(L(1)\) |
\(\approx\) |
\(0.5020173180 - 0.6264198274i\) |
\(L(1)\) |
\(\approx\) |
\(0.5020173180 - 0.6264198274i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 283 | \( 1 \) |
good | 2 | \( 1 + (0.695 - 0.718i)T \) |
| 3 | \( 1 + (-0.538 - 0.842i)T \) |
| 5 | \( 1 + (-0.420 - 0.907i)T \) |
| 7 | \( 1 + (-0.645 + 0.763i)T \) |
| 11 | \( 1 + (-0.0334 + 0.999i)T \) |
| 13 | \( 1 + (-0.979 - 0.199i)T \) |
| 17 | \( 1 + (0.100 - 0.994i)T \) |
| 19 | \( 1 + (-0.979 - 0.199i)T \) |
| 23 | \( 1 + (0.964 - 0.264i)T \) |
| 29 | \( 1 + (0.359 - 0.933i)T \) |
| 31 | \( 1 + (-0.892 + 0.451i)T \) |
| 37 | \( 1 + (0.860 - 0.509i)T \) |
| 41 | \( 1 + (-0.741 + 0.670i)T \) |
| 43 | \( 1 + (0.480 - 0.876i)T \) |
| 47 | \( 1 + (-0.979 + 0.199i)T \) |
| 53 | \( 1 + (-0.997 - 0.0667i)T \) |
| 59 | \( 1 + (-0.166 - 0.986i)T \) |
| 61 | \( 1 + (-0.944 + 0.328i)T \) |
| 67 | \( 1 + (-0.824 - 0.565i)T \) |
| 71 | \( 1 + (-0.0334 + 0.999i)T \) |
| 73 | \( 1 + (-0.892 - 0.451i)T \) |
| 79 | \( 1 + (0.480 + 0.876i)T \) |
| 83 | \( 1 + (0.991 - 0.133i)T \) |
| 89 | \( 1 + (0.480 - 0.876i)T \) |
| 97 | \( 1 + (0.964 - 0.264i)T \) |
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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
−26.22201838623592314514986207221, −25.53489519440456144058846466312, −23.91838846114229556306000680593, −23.498354946777571928672743358814, −22.58444017305336465237447659435, −21.897786754742506100954069863148, −21.29366525906135093978714113056, −19.92394597758250565162852412250, −18.925996905635139705518449071894, −17.52015945656785141598281059872, −16.7425455526360542014412636949, −16.154637305901373096753379912707, −14.948866070450021013774355939950, −14.60708444974997249044461763936, −13.3159120316082901295526260220, −12.24949055842391153604224981687, −11.113618436962372365766193156119, −10.449738696041613689803604179597, −9.11040038723053084540682935939, −7.7886922695219428636067519997, −6.68098454112020529740802998269, −6.04054749901502754509976751962, −4.69478226726303724838527120629, −3.71323463455339460946849207302, −3.02017019233999039025755064717,
0.264950753198858125567169302722, 1.85542244935573115940086979245, 2.84007814030629029584033331121, 4.60975150196056052048592409454, 5.20535646913725573560864355370, 6.396879526894169327502050661706, 7.4643259209550613843777054793, 8.952331034514552050011095146215, 9.88467221497024842250261110042, 11.260697897253116480704967698862, 12.170596555008221533980963676918, 12.61826301858971202518063480895, 13.263809865295609751857584538518, 14.681258957600281338340875419723, 15.60261874870920797799650731635, 16.706839729758194232906518354739, 17.79730274387917718485658557457, 18.86460156434509663212339291023, 19.54840967894852659024640563089, 20.29082509126052667933959127507, 21.42539873948386936987498914177, 22.44630626173050465716294089831, 23.08745421063322821507704931917, 23.82584406740385437157260848743, 24.959793757388657337039665804804