L(s) = 1 | + (−0.979 + 0.200i)2-s + (0.632 − 0.774i)3-s + (0.919 − 0.391i)4-s + (0.960 + 0.278i)5-s + (−0.464 + 0.885i)6-s + (−0.822 + 0.568i)8-s + (−0.200 − 0.979i)9-s + (−0.996 − 0.0804i)10-s + (−0.979 − 0.200i)11-s + (0.278 − 0.960i)12-s + (0.822 − 0.568i)15-s + (0.692 − 0.721i)16-s + (0.428 − 0.903i)17-s + (0.391 + 0.919i)18-s + (−0.866 − 0.5i)19-s + (0.992 − 0.120i)20-s + ⋯ |
L(s) = 1 | + (−0.979 + 0.200i)2-s + (0.632 − 0.774i)3-s + (0.919 − 0.391i)4-s + (0.960 + 0.278i)5-s + (−0.464 + 0.885i)6-s + (−0.822 + 0.568i)8-s + (−0.200 − 0.979i)9-s + (−0.996 − 0.0804i)10-s + (−0.979 − 0.200i)11-s + (0.278 − 0.960i)12-s + (0.822 − 0.568i)15-s + (0.692 − 0.721i)16-s + (0.428 − 0.903i)17-s + (0.391 + 0.919i)18-s + (−0.866 − 0.5i)19-s + (0.992 − 0.120i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6935178942 - 0.9351816256i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6935178942 - 0.9351816256i\) |
\(L(1)\) |
\(\approx\) |
\(0.8655250856 - 0.3114029302i\) |
\(L(1)\) |
\(\approx\) |
\(0.8655250856 - 0.3114029302i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.979 + 0.200i)T \) |
| 3 | \( 1 + (0.632 - 0.774i)T \) |
| 5 | \( 1 + (0.960 + 0.278i)T \) |
| 11 | \( 1 + (-0.979 - 0.200i)T \) |
| 17 | \( 1 + (0.428 - 0.903i)T \) |
| 19 | \( 1 + (-0.866 - 0.5i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.748 + 0.663i)T \) |
| 31 | \( 1 + (0.999 - 0.0402i)T \) |
| 37 | \( 1 + (-0.534 - 0.845i)T \) |
| 41 | \( 1 + (0.935 - 0.354i)T \) |
| 43 | \( 1 + (-0.885 + 0.464i)T \) |
| 47 | \( 1 + (-0.391 + 0.919i)T \) |
| 53 | \( 1 + (0.428 - 0.903i)T \) |
| 59 | \( 1 + (0.721 - 0.692i)T \) |
| 61 | \( 1 + (-0.428 - 0.903i)T \) |
| 67 | \( 1 + (-0.600 - 0.799i)T \) |
| 71 | \( 1 + (-0.935 + 0.354i)T \) |
| 73 | \( 1 + (0.979 + 0.200i)T \) |
| 79 | \( 1 + (-0.919 - 0.391i)T \) |
| 83 | \( 1 + (-0.935 - 0.354i)T \) |
| 89 | \( 1 + (0.866 + 0.5i)T \) |
| 97 | \( 1 + (0.239 - 0.970i)T \) |
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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
−21.19433133019909897127737265404, −20.82866234253791159573677342559, −19.94135871534260003228423207192, −19.136959524554800451948946995661, −18.493380145382154816614693972, −17.4353256966020494875803584074, −16.95487338899008848650601323001, −16.21889775883652162823100536793, −15.2542340984883656452459815752, −14.82509088589756282070674580210, −13.52086147043330250731438938496, −13.013266472392860343199800941555, −11.90780610038894958401879967044, −10.74531539019166075238159437296, −10.21060822163395876620964112634, −9.7330686332708956838934098285, −8.75805979064862248749430452688, −8.24103932394454415839677563291, −7.3477738646276560913875958360, −6.10237967638132766521829609020, −5.348530729422678651032259760625, −4.15711502851017713264950732278, −3.0583583159386590296778174777, −2.26747940649514136494282342631, −1.446139843677075326059020063656,
0.55986663724272506530532089017, 1.74791462609236293522334673024, 2.53144164107653881390652973720, 3.11694142947119225236687302406, 5.0179883836224062288241121152, 5.974287006742397045367786463573, 6.73805992947587597162784634744, 7.402981705246080710116300364493, 8.3041943043267174428436398064, 9.00967068692642053407213238602, 9.7447016825941111649943579552, 10.58364322449551493662438235141, 11.3522060262134577984759972408, 12.55509575077238663144398609714, 13.16531689911726895185481990805, 14.165729148458211692218005378865, 14.70012150631600637084612128713, 15.62762593932271939478598554415, 16.54008938320958263830721491775, 17.43699046381188508901040344649, 18.01240048322287308157736882997, 18.65655919795795181537797702660, 19.15073710345918282474527968825, 20.11718383899470900726811397542, 21.003085254014372548313344882678