L(s) = 1 | + (0.991 + 0.133i)2-s + (−0.166 − 0.986i)3-s + (0.964 + 0.264i)4-s + (−0.944 − 0.328i)5-s + (−0.0334 − 0.999i)6-s + (0.784 + 0.619i)7-s + (0.920 + 0.390i)8-s + (−0.944 + 0.328i)9-s + (−0.892 − 0.451i)10-s + (0.964 − 0.264i)11-s + (0.100 − 0.994i)12-s + (−0.0334 − 0.999i)13-s + (0.695 + 0.718i)14-s + (−0.166 + 0.986i)15-s + (0.860 + 0.509i)16-s + (0.695 − 0.718i)17-s + ⋯ |
L(s) = 1 | + (0.991 + 0.133i)2-s + (−0.166 − 0.986i)3-s + (0.964 + 0.264i)4-s + (−0.944 − 0.328i)5-s + (−0.0334 − 0.999i)6-s + (0.784 + 0.619i)7-s + (0.920 + 0.390i)8-s + (−0.944 + 0.328i)9-s + (−0.892 − 0.451i)10-s + (0.964 − 0.264i)11-s + (0.100 − 0.994i)12-s + (−0.0334 − 0.999i)13-s + (0.695 + 0.718i)14-s + (−0.166 + 0.986i)15-s + (0.860 + 0.509i)16-s + (0.695 − 0.718i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.859285540 - 0.9129446923i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.859285540 - 0.9129446923i\) |
\(L(1)\) |
\(\approx\) |
\(1.649766562 - 0.4559904245i\) |
\(L(1)\) |
\(\approx\) |
\(1.649766562 - 0.4559904245i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 283 | \( 1 \) |
good | 2 | \( 1 + (0.991 + 0.133i)T \) |
| 3 | \( 1 + (-0.166 - 0.986i)T \) |
| 5 | \( 1 + (-0.944 - 0.328i)T \) |
| 7 | \( 1 + (0.784 + 0.619i)T \) |
| 11 | \( 1 + (0.964 - 0.264i)T \) |
| 13 | \( 1 + (-0.0334 - 0.999i)T \) |
| 17 | \( 1 + (0.695 - 0.718i)T \) |
| 19 | \( 1 + (-0.0334 - 0.999i)T \) |
| 23 | \( 1 + (-0.538 + 0.842i)T \) |
| 29 | \( 1 + (-0.979 - 0.199i)T \) |
| 31 | \( 1 + (-0.824 - 0.565i)T \) |
| 37 | \( 1 + (-0.420 - 0.907i)T \) |
| 41 | \( 1 + (0.920 - 0.390i)T \) |
| 43 | \( 1 + (-0.645 + 0.763i)T \) |
| 47 | \( 1 + (-0.0334 + 0.999i)T \) |
| 53 | \( 1 + (0.860 - 0.509i)T \) |
| 59 | \( 1 + (0.231 + 0.972i)T \) |
| 61 | \( 1 + (-0.892 + 0.451i)T \) |
| 67 | \( 1 + (0.100 + 0.994i)T \) |
| 71 | \( 1 + (0.964 - 0.264i)T \) |
| 73 | \( 1 + (-0.824 + 0.565i)T \) |
| 79 | \( 1 + (-0.645 - 0.763i)T \) |
| 83 | \( 1 + (0.480 + 0.876i)T \) |
| 89 | \( 1 + (-0.645 + 0.763i)T \) |
| 97 | \( 1 + (-0.538 + 0.842i)T \) |
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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
−25.80379929039251983012585060570, −24.4831764884129577258391468324, −23.60219636754165213948595563660, −22.99372761992116670778245246977, −22.12221305473370034603591148373, −21.31760059209380460737059132329, −20.386504134002260281223754739229, −19.81484907055884937034431956934, −18.63401631307911321810147155959, −16.794441796145104674615735594548, −16.59734614523842350953902745878, −15.27810296521629444451411028685, −14.503498098168070398038448546758, −14.177671314161056585571112838812, −12.327469462269138431481481234185, −11.67191044894906512518806943176, −10.87875666732682667546494045342, −10.04431413302860798730058344584, −8.48870156163914001630440566258, −7.30444003029794615497743009825, −6.222143173074189090627502755721, −4.89144211869886444418759127124, −3.99363103659724860680468177204, −3.58723846220666855853044043693, −1.719065018103525842516193828964,
1.216822381686064855877959561277, 2.60361716176174830792305842332, 3.80079027227286569439535946856, 5.15564451608428225615631865726, 5.86786570128502349396694367138, 7.30687099755688993857236111883, 7.79248730133123655252884770908, 8.946517093060152433731910001012, 11.136989249030651438982312070395, 11.59743504504022460999490936821, 12.36304800135015403516757207198, 13.221435193149882423042841748848, 14.341899047883419028433094756419, 15.059640196663319911741585288872, 16.10385033887506589621821996462, 17.12845302545121350750197226806, 18.109315200049849752520745086973, 19.35795193272833088985233888022, 19.95291527929892940197524557470, 20.91241299329272437044232903890, 22.17447963170472593494446071848, 22.82295521169256946675983443858, 23.76646897676388757195213456838, 24.44670039185548402133964732313, 24.91516394730324736234477626187