L(s) = 1 | + (0.939 − 0.342i)2-s + (0.939 + 0.342i)3-s + (0.766 − 0.642i)4-s + 6-s + (−0.173 − 0.984i)7-s + (0.5 − 0.866i)8-s + (0.766 + 0.642i)9-s + (−0.5 + 0.866i)11-s + (0.939 − 0.342i)12-s + (−0.766 + 0.642i)13-s + (−0.5 − 0.866i)14-s + (0.173 − 0.984i)16-s + (−0.766 − 0.642i)17-s + (0.939 + 0.342i)18-s + (−0.939 − 0.342i)19-s + ⋯ |
L(s) = 1 | + (0.939 − 0.342i)2-s + (0.939 + 0.342i)3-s + (0.766 − 0.642i)4-s + 6-s + (−0.173 − 0.984i)7-s + (0.5 − 0.866i)8-s + (0.766 + 0.642i)9-s + (−0.5 + 0.866i)11-s + (0.939 − 0.342i)12-s + (−0.766 + 0.642i)13-s + (−0.5 − 0.866i)14-s + (0.173 − 0.984i)16-s + (−0.766 − 0.642i)17-s + (0.939 + 0.342i)18-s + (−0.939 − 0.342i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.402278691 - 0.5979289452i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.402278691 - 0.5979289452i\) |
\(L(1)\) |
\(\approx\) |
\(2.091884467 - 0.3675858007i\) |
\(L(1)\) |
\(\approx\) |
\(2.091884467 - 0.3675858007i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 2 | \( 1 + (0.939 - 0.342i)T \) |
| 3 | \( 1 + (0.939 + 0.342i)T \) |
| 7 | \( 1 + (-0.173 - 0.984i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.766 + 0.642i)T \) |
| 17 | \( 1 + (-0.766 - 0.642i)T \) |
| 19 | \( 1 + (-0.939 - 0.342i)T \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + T \) |
| 41 | \( 1 + (0.766 - 0.642i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (0.5 + 0.866i)T \) |
| 53 | \( 1 + (-0.173 + 0.984i)T \) |
| 59 | \( 1 + (0.173 - 0.984i)T \) |
| 61 | \( 1 + (0.766 - 0.642i)T \) |
| 67 | \( 1 + (-0.173 - 0.984i)T \) |
| 71 | \( 1 + (-0.939 - 0.342i)T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + (0.173 + 0.984i)T \) |
| 83 | \( 1 + (-0.766 - 0.642i)T \) |
| 89 | \( 1 + (0.173 - 0.984i)T \) |
| 97 | \( 1 + (0.5 + 0.866i)T \) |
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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
−26.84735345203757100528943741289, −26.14944341498812569258057578074, −24.97641482820052694387214612945, −24.66733254255554576524818317583, −23.67694071705839862497725905623, −22.44095055471852501708471172406, −21.517814074484874124061920711485, −20.832064594377349558730494825015, −19.62351553865699539889566749676, −18.84901002368527095098925248927, −17.53231443592295565557746374970, −16.1997525646825406916826175183, −15.11987483970539187560434611014, −14.7565614636768930065449583273, −13.33187265745503476631745719604, −12.84234960980864560173369768819, −11.767499557321628536722274567720, −10.31992390344944262994141374629, −8.66952142849258077889815956651, −8.07591320034656261172697901174, −6.69956573679099638333420097154, −5.71845780182986480272105669289, −4.31384696525372621672174598638, −2.95039963479978361839894309927, −2.24700451696187811317899161108,
1.85944921225944632917001584369, 2.94657147940939088929702382022, 4.2464997410042108158203965983, 4.85917552376657894539692425593, 6.83357076035440828438900680674, 7.504852304517822226611831594266, 9.27810962562250824488533137294, 10.174631481198949279568874778933, 11.13074954334971553065752115973, 12.61956503244951972518909926885, 13.42010486430302576477648580829, 14.25091472421028831023415849065, 15.16785160197026986274794368303, 16.005873483323317095982485442005, 17.26831833379825397891926519161, 18.95099674866401329646924140080, 19.789334067501172418304132060994, 20.44761384073971520804667436055, 21.29532308844993650865861805624, 22.24789233004131336629505996240, 23.31886811041480395967792992390, 24.142715137122035695882593523972, 25.21605987114781439089530189427, 26.07448179550257105169996792684, 27.02291053883934229937500801753