L(s) = 1 | + (−0.207 + 0.978i)2-s + (−0.913 − 0.406i)4-s + (−0.743 + 0.669i)5-s + (0.587 + 0.809i)7-s + (0.587 − 0.809i)8-s + (−0.5 − 0.866i)10-s + (−0.913 + 0.406i)14-s + (0.669 + 0.743i)16-s + (−0.669 − 0.743i)17-s + (0.406 + 0.913i)19-s + (0.951 − 0.309i)20-s + 23-s + (0.104 − 0.994i)25-s + (−0.207 − 0.978i)28-s + (−0.104 − 0.994i)29-s + ⋯ |
L(s) = 1 | + (−0.207 + 0.978i)2-s + (−0.913 − 0.406i)4-s + (−0.743 + 0.669i)5-s + (0.587 + 0.809i)7-s + (0.587 − 0.809i)8-s + (−0.5 − 0.866i)10-s + (−0.913 + 0.406i)14-s + (0.669 + 0.743i)16-s + (−0.669 − 0.743i)17-s + (0.406 + 0.913i)19-s + (0.951 − 0.309i)20-s + 23-s + (0.104 − 0.994i)25-s + (−0.207 − 0.978i)28-s + (−0.104 − 0.994i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.3829929106 + 0.6398886923i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.3829929106 + 0.6398886923i\) |
\(L(1)\) |
\(\approx\) |
\(0.5683052853 + 0.5177706496i\) |
\(L(1)\) |
\(\approx\) |
\(0.5683052853 + 0.5177706496i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.207 + 0.978i)T \) |
| 5 | \( 1 + (-0.743 + 0.669i)T \) |
| 7 | \( 1 + (0.587 + 0.809i)T \) |
| 17 | \( 1 + (-0.669 - 0.743i)T \) |
| 19 | \( 1 + (0.406 + 0.913i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (-0.104 - 0.994i)T \) |
| 31 | \( 1 + (-0.207 + 0.978i)T \) |
| 37 | \( 1 + (-0.406 + 0.913i)T \) |
| 41 | \( 1 + (-0.587 + 0.809i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (0.994 + 0.104i)T \) |
| 53 | \( 1 + (-0.309 - 0.951i)T \) |
| 59 | \( 1 + (0.406 - 0.913i)T \) |
| 61 | \( 1 + (-0.309 + 0.951i)T \) |
| 67 | \( 1 - iT \) |
| 71 | \( 1 + (-0.743 + 0.669i)T \) |
| 73 | \( 1 + (0.587 + 0.809i)T \) |
| 79 | \( 1 + (-0.669 + 0.743i)T \) |
| 83 | \( 1 + (0.207 + 0.978i)T \) |
| 89 | \( 1 + (-0.866 + 0.5i)T \) |
| 97 | \( 1 + (0.951 - 0.309i)T \) |
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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
−20.37261318792874618522251265266, −19.63241098989040558218117293496, −19.0750365696565282478122419932, −18.047294012080283820921524916098, −17.26772325270560901496837535443, −16.82538436207327517512380605383, −15.747874551252361761062658376675, −14.85920403777695018250670028723, −13.89817997505707114664706957267, −13.14073870829588320981016372531, −12.54323096996749467644980664998, −11.597119664649327366769252730373, −10.9869683076628166243369212827, −10.42555894515484957004941168890, −9.05186827372051370804420262582, −8.82438949134687636922587204377, −7.67877382585088159679491099717, −7.17351137794120456051535112402, −5.49829828777615113153617697813, −4.612494090713663444615148078575, −4.080999893288878760241547328861, −3.17260229931685111003711534983, −1.925089085422644713045357922954, −0.99288667397919990157470843196, −0.20363909521880983886933391808,
1.136341581218282175848385685724, 2.517522738557390749750610127044, 3.5962351877563285354875877311, 4.62295728426252772154106858729, 5.338167420227114072184845081051, 6.33643787490311843514044786180, 7.086163081341540407704965489889, 7.87956232598259645742879530849, 8.53113587898648001002597915753, 9.35520780248130618112486014254, 10.31588373244440816206223703198, 11.24869426180550149495244845800, 11.95570686321217578534789258784, 12.93631386345352403555704639518, 14.02928638518302386221638432759, 14.52009271865083154242348041757, 15.44133796335486422297555068020, 15.66799941918352884373044793039, 16.67917892064941896530591957030, 17.57938853724094305525484873497, 18.31215449711836879282956399762, 18.78043244374140850760788439939, 19.49396294404532081284451320068, 20.55346927731371081860738474775, 21.49656364287478911393028081235