L(s) = 1 | + (0.5 + 0.866i)7-s + (−0.5 + 0.866i)11-s + (−0.766 + 0.642i)13-s + (0.173 − 0.984i)17-s + (−0.939 − 0.342i)23-s + (−0.173 − 0.984i)29-s + (0.5 + 0.866i)31-s − 37-s + (0.766 + 0.642i)41-s + (−0.939 + 0.342i)43-s + (0.173 + 0.984i)47-s + (−0.5 + 0.866i)49-s + (−0.939 − 0.342i)53-s + (−0.173 + 0.984i)59-s + (0.939 + 0.342i)61-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)7-s + (−0.5 + 0.866i)11-s + (−0.766 + 0.642i)13-s + (0.173 − 0.984i)17-s + (−0.939 − 0.342i)23-s + (−0.173 − 0.984i)29-s + (0.5 + 0.866i)31-s − 37-s + (0.766 + 0.642i)41-s + (−0.939 + 0.342i)43-s + (0.173 + 0.984i)47-s + (−0.5 + 0.866i)49-s + (−0.939 − 0.342i)53-s + (−0.173 + 0.984i)59-s + (0.939 + 0.342i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.000652i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.000652i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.0001462463577 + 0.4481250490i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0001462463577 + 0.4481250490i\) |
\(L(1)\) |
\(\approx\) |
\(0.8364673002 + 0.1940814634i\) |
\(L(1)\) |
\(\approx\) |
\(0.8364673002 + 0.1940814634i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 7 | \( 1 + (0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.766 + 0.642i)T \) |
| 17 | \( 1 + (0.173 - 0.984i)T \) |
| 23 | \( 1 + (-0.939 - 0.342i)T \) |
| 29 | \( 1 + (-0.173 - 0.984i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + (0.766 + 0.642i)T \) |
| 43 | \( 1 + (-0.939 + 0.342i)T \) |
| 47 | \( 1 + (0.173 + 0.984i)T \) |
| 53 | \( 1 + (-0.939 - 0.342i)T \) |
| 59 | \( 1 + (-0.173 + 0.984i)T \) |
| 61 | \( 1 + (0.939 + 0.342i)T \) |
| 67 | \( 1 + (-0.173 - 0.984i)T \) |
| 71 | \( 1 + (-0.939 + 0.342i)T \) |
| 73 | \( 1 + (-0.766 - 0.642i)T \) |
| 79 | \( 1 + (-0.766 - 0.642i)T \) |
| 83 | \( 1 + (0.5 + 0.866i)T \) |
| 89 | \( 1 + (0.766 - 0.642i)T \) |
| 97 | \( 1 + (0.173 - 0.984i)T \) |
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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
−19.25458464317587583927607348015, −18.696920004516854566086345066518, −17.59146401044600709394429296711, −17.34258472334096859448900147919, −16.45113936692234308482245266342, −15.765185537088641395104265089649, −14.87100887403369171181352879158, −14.26201048402898819728852339039, −13.513523441819791274002619908289, −12.862963118976358999193307398230, −11.99992856767814023352935291044, −11.17920881384614870916929191724, −10.4074807454220098083066587567, −10.04719746661277656822190431261, −8.824403020342323889552880577423, −8.06172844563664415628051591155, −7.57451884497143610609408511011, −6.63118605264704025277604868737, −5.66017209166230780735359256, −5.0533004127885411969777073676, −3.99672840938045513295647444909, −3.37030392132688862434736254882, −2.27323173580326298910178575347, −1.30505737570784822655482158854, −0.14096170013169358009200394205,
1.54347320868836884191678642470, 2.32125867857619535069717683891, 2.99425198177722605530789148877, 4.4191932446070159860785569143, 4.82865823042247655806750107921, 5.7057847502727486265019838306, 6.6201366842663039954674718918, 7.48674069068610566845413409931, 8.10403712613567856048979738020, 9.05821888132947919307825886661, 9.70620412523435493252001746806, 10.39084329843544809619599498904, 11.512385788139567469852817246082, 11.98457674463448008403890337832, 12.57867199762948645244072813073, 13.57712369274960907760394787743, 14.377548138260478124387605381752, 14.89287602380199890226880196842, 15.77388201588092355990285703781, 16.24147849670615373011811275737, 17.37667113542381296076175691463, 17.83167159042594672920599165310, 18.55424028906619416624264966319, 19.21814405083734468632939794392, 20.03396590740158559996294457384