L(s) = 1 | − i·2-s − 4-s − i·7-s + i·8-s − i·11-s − 14-s + 16-s − 17-s − i·19-s − 22-s − 23-s + i·28-s − 29-s − i·31-s − i·32-s + ⋯ |
L(s) = 1 | − i·2-s − 4-s − i·7-s + i·8-s − i·11-s − 14-s + 16-s − 17-s − i·19-s − 22-s − 23-s + i·28-s − 29-s − i·31-s − i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1183735598 - 0.7994500432i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1183735598 - 0.7994500432i\) |
\(L(1)\) |
\(\approx\) |
\(0.6142942290 - 0.5916149716i\) |
\(L(1)\) |
\(\approx\) |
\(0.6142942290 - 0.5916149716i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 \) |
| 19 | \( 1 - iT \) |
| 23 | \( 1 + iT \) |
| 29 | \( 1 \) |
| 31 | \( 1 \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 \) |
| 43 | \( 1 \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 \) |
| 71 | \( 1 - iT \) |
| 73 | \( 1 \) |
| 79 | \( 1 \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 \) |
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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
−27.41006243987611155963671540654, −26.23200205503814086363158546972, −25.46974604796932136141303764925, −24.69185414663387701958618302826, −23.84147662800606836924583005109, −22.65538483279558990330550147423, −22.12933784535157338354543743263, −20.9098484236809140232681137912, −19.60489772663022005346137826126, −18.414810051807352470480567833, −17.8641124960387946142430584313, −16.72519199332928960599423635461, −15.6430610809627454211536287917, −15.03312500245828279683798262155, −13.99932041331792509897496875416, −12.7875809908972514332308420575, −11.96825796772416932360022795304, −10.256544325800496919111887148415, −9.224087961347064058563092687740, −8.31270946578051809490057019791, −7.16140482539580902988353550661, −6.06963261297330422930417170921, −5.074291092357082985754028130683, −3.86657144658667160273343369276, −2.04126972779369719503925112940,
0.63009750646815891430198310668, 2.25593206268667393946227195212, 3.62244400894268124842017547758, 4.52782902890317638678781266719, 5.98584635364490389199387102701, 7.530822725444980359488024397958, 8.72051968651867390605263620428, 9.7516689282910421104382374259, 10.93198526329444982405328910280, 11.41268235140230869157901562984, 12.95255242439687484454914273737, 13.56754764567687374460655584569, 14.51746027543506212886247753767, 16.038298898148479165301523950196, 17.17064299636546259835292021488, 17.9981227256340156220179718837, 19.15838172825543823491155155481, 19.92366635228757798276734725896, 20.71809362880821641175602833471, 21.81756224779155642958134917089, 22.50706789048203771526511897538, 23.69323459756635844169686410293, 24.339907653884565881275096629752, 26.20205873205330436406302720953, 26.53970163819884286869601109403