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
−27.02291053883934229937500801753, −26.07448179550257105169996792684, −25.21605987114781439089530189427, −24.142715137122035695882593523972, −23.31886811041480395967792992390, −22.24789233004131336629505996240, −21.29532308844993650865861805624, −20.44761384073971520804667436055, −19.789334067501172418304132060994, −18.95099674866401329646924140080, −17.26831833379825397891926519161, −16.005873483323317095982485442005, −15.16785160197026986274794368303, −14.25091472421028831023415849065, −13.42010486430302576477648580829, −12.61956503244951972518909926885, −11.13074954334971553065752115973, −10.174631481198949279568874778933, −9.27810962562250824488533137294, −7.504852304517822226611831594266, −6.83357076035440828438900680674, −4.85917552376657894539692425593, −4.2464997410042108158203965983, −2.94657147940939088929702382022, −1.85944921225944632917001584369,
2.24700451696187811317899161108, 2.95039963479978361839894309927, 4.31384696525372621672174598638, 5.71845780182986480272105669289, 6.69956573679099638333420097154, 8.07591320034656261172697901174, 8.66952142849258077889815956651, 10.31992390344944262994141374629, 11.767499557321628536722274567720, 12.84234960980864560173369768819, 13.33187265745503476631745719604, 14.7565614636768930065449583273, 15.11987483970539187560434611014, 16.1997525646825406916826175183, 17.53231443592295565557746374970, 18.84901002368527095098925248927, 19.62351553865699539889566749676, 20.832064594377349558730494825015, 21.517814074484874124061920711485, 22.44095055471852501708471172406, 23.67694071705839862497725905623, 24.66733254255554576524818317583, 24.97641482820052694387214612945, 26.14944341498812569258057578074, 26.84735345203757100528943741289