L(s) = 1 | + (−0.729 − 0.683i)3-s + (−0.412 + 0.910i)5-s + (0.0654 + 0.997i)9-s + (−0.528 + 0.849i)11-s + (−0.0980 + 0.995i)13-s + (0.923 − 0.382i)15-s + (−0.130 − 0.991i)17-s + (−0.986 + 0.162i)19-s + (−0.751 − 0.659i)23-s + (−0.659 − 0.751i)25-s + (0.634 − 0.773i)27-s + (−0.881 − 0.471i)29-s + (−0.965 − 0.258i)31-s + (0.965 − 0.258i)33-s + (−0.935 + 0.352i)37-s + ⋯ |
L(s) = 1 | + (−0.729 − 0.683i)3-s + (−0.412 + 0.910i)5-s + (0.0654 + 0.997i)9-s + (−0.528 + 0.849i)11-s + (−0.0980 + 0.995i)13-s + (0.923 − 0.382i)15-s + (−0.130 − 0.991i)17-s + (−0.986 + 0.162i)19-s + (−0.751 − 0.659i)23-s + (−0.659 − 0.751i)25-s + (0.634 − 0.773i)27-s + (−0.881 − 0.471i)29-s + (−0.965 − 0.258i)31-s + (0.965 − 0.258i)33-s + (−0.935 + 0.352i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.416 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.416 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4080021490 - 0.2619024919i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4080021490 - 0.2619024919i\) |
\(L(1)\) |
\(\approx\) |
\(0.6213160314 + 0.004899936354i\) |
\(L(1)\) |
\(\approx\) |
\(0.6213160314 + 0.004899936354i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.729 - 0.683i)T \) |
| 5 | \( 1 + (-0.412 + 0.910i)T \) |
| 11 | \( 1 + (-0.528 + 0.849i)T \) |
| 13 | \( 1 + (-0.0980 + 0.995i)T \) |
| 17 | \( 1 + (-0.130 - 0.991i)T \) |
| 19 | \( 1 + (-0.986 + 0.162i)T \) |
| 23 | \( 1 + (-0.751 - 0.659i)T \) |
| 29 | \( 1 + (-0.881 - 0.471i)T \) |
| 31 | \( 1 + (-0.965 - 0.258i)T \) |
| 37 | \( 1 + (-0.935 + 0.352i)T \) |
| 41 | \( 1 + (-0.980 - 0.195i)T \) |
| 43 | \( 1 + (0.956 + 0.290i)T \) |
| 47 | \( 1 + (-0.608 - 0.793i)T \) |
| 53 | \( 1 + (0.849 + 0.528i)T \) |
| 59 | \( 1 + (0.812 - 0.582i)T \) |
| 61 | \( 1 + (0.973 + 0.227i)T \) |
| 67 | \( 1 + (0.683 - 0.729i)T \) |
| 71 | \( 1 + (0.831 + 0.555i)T \) |
| 73 | \( 1 + (-0.442 + 0.896i)T \) |
| 79 | \( 1 + (0.991 + 0.130i)T \) |
| 83 | \( 1 + (0.773 - 0.634i)T \) |
| 89 | \( 1 + (0.946 + 0.321i)T \) |
| 97 | \( 1 + (0.707 - 0.707i)T \) |
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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
−20.46390214370278446847024186482, −19.68531434193324953533517482460, −18.95257353939635365788880679303, −17.832528634170978575342183059611, −17.373050451683367899355749704, −16.51929637922822569049450551246, −16.06528349765789339765502144888, −15.30418208566360529049533097825, −14.729519644800835785565039182909, −13.40909652692376626803533559303, −12.769959975234229218428573932583, −12.19049049895310020010793527797, −11.18554873687686554100758908784, −10.70279093225284237442203874117, −9.90889694512574203562890053160, −8.8932765360946059951170903308, −8.371176863864755023208115584681, −7.462058733391666516394322372929, −6.24755685721285689259906176538, −5.4973644708081025863212185877, −5.04984999418447746076347394527, −3.86430683719019605498611462947, −3.511945266203636332336282143737, −1.95167426658556145497226466950, −0.71025137340766597128493754916,
0.27472959424268957322974265779, 2.05688634423509319435931878977, 2.25654360405705630242946528298, 3.73277136781699520841928765631, 4.57442032710419306995382784727, 5.46621022813986505225899651909, 6.49997719605948764017113795454, 6.98432038418198921026098466837, 7.60654302739775476549642047751, 8.50142519279759496998770475239, 9.72218801213821328498112460341, 10.434959888344065601116838952953, 11.19958737959374222090127810419, 11.80820954033164142235848477895, 12.47747940876830262913611934568, 13.33210424105346962283804687906, 14.13361005504943042254743394813, 14.8446136005765230221560829442, 15.71715190981921006132930271494, 16.44760938025819081861329835606, 17.2206713408276262788235665458, 18.03642947947885078189887189217, 18.60706660502151289683775973042, 19.023952281802650732228133646811, 19.94778489552268465815586797716