L(s) = 1 | + (0.743 − 0.669i)2-s + (0.258 − 0.965i)3-s + (0.104 − 0.994i)4-s + (0.406 + 0.913i)5-s + (−0.453 − 0.891i)6-s + (−0.587 − 0.809i)8-s + (−0.866 − 0.5i)9-s + (0.913 + 0.406i)10-s + (−0.358 − 0.933i)11-s + (−0.933 − 0.358i)12-s + (0.891 − 0.453i)13-s + (0.987 − 0.156i)15-s + (−0.978 − 0.207i)16-s + (0.933 − 0.358i)17-s + (−0.978 + 0.207i)18-s + (−0.838 + 0.544i)19-s + ⋯ |
L(s) = 1 | + (0.743 − 0.669i)2-s + (0.258 − 0.965i)3-s + (0.104 − 0.994i)4-s + (0.406 + 0.913i)5-s + (−0.453 − 0.891i)6-s + (−0.587 − 0.809i)8-s + (−0.866 − 0.5i)9-s + (0.913 + 0.406i)10-s + (−0.358 − 0.933i)11-s + (−0.933 − 0.358i)12-s + (0.891 − 0.453i)13-s + (0.987 − 0.156i)15-s + (−0.978 − 0.207i)16-s + (0.933 − 0.358i)17-s + (−0.978 + 0.207i)18-s + (−0.838 + 0.544i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 + 0.0806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 + 0.0806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1069659058 - 2.648131507i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1069659058 - 2.648131507i\) |
\(L(1)\) |
\(\approx\) |
\(1.095466001 - 1.224768986i\) |
\(L(1)\) |
\(\approx\) |
\(1.095466001 - 1.224768986i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 41 | \( 1 \) |
good | 2 | \( 1 + (0.743 - 0.669i)T \) |
| 3 | \( 1 + (0.258 - 0.965i)T \) |
| 5 | \( 1 + (0.406 + 0.913i)T \) |
| 11 | \( 1 + (-0.358 - 0.933i)T \) |
| 13 | \( 1 + (0.891 - 0.453i)T \) |
| 17 | \( 1 + (0.933 - 0.358i)T \) |
| 19 | \( 1 + (-0.838 + 0.544i)T \) |
| 23 | \( 1 + (-0.669 - 0.743i)T \) |
| 29 | \( 1 + (-0.156 - 0.987i)T \) |
| 31 | \( 1 + (-0.913 - 0.406i)T \) |
| 37 | \( 1 + (0.913 - 0.406i)T \) |
| 43 | \( 1 + (-0.951 - 0.309i)T \) |
| 47 | \( 1 + (0.998 + 0.0523i)T \) |
| 53 | \( 1 + (0.777 - 0.629i)T \) |
| 59 | \( 1 + (-0.978 + 0.207i)T \) |
| 61 | \( 1 + (-0.207 + 0.978i)T \) |
| 67 | \( 1 + (0.629 + 0.777i)T \) |
| 71 | \( 1 + (-0.987 - 0.156i)T \) |
| 73 | \( 1 + (-0.866 + 0.5i)T \) |
| 79 | \( 1 + (0.965 - 0.258i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (-0.544 - 0.838i)T \) |
| 97 | \( 1 + (0.987 - 0.156i)T \) |
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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
−25.61956149864665811647705418408, −25.1528689288362131561811260776, −23.61679511024975912902745553922, −23.3962628840196454529596109708, −21.89415404103299505868177729015, −21.456307527695432028778299340727, −20.58384088092683519471209810199, −19.94365122977399920564971713561, −18.15234933699081536572243690356, −17.07903989462276055188018740112, −16.4587250356389267470991382628, −15.63051296963775848900834829856, −14.7978178176007735521287691847, −13.82982801431737959237710026310, −12.96378185079018716883837835804, −11.99941961412568758158437747065, −10.72127361614009774553513527875, −9.471539226792544396295530656517, −8.669315650685925554315195205589, −7.69635048007645103789866867924, −6.17280553705115132823578593372, −5.23200310948340869671775045723, −4.435292527203919564977643703352, −3.47733248737223558727212467354, −1.96052126381836815590987319167,
0.56286357793997310403033622478, 1.939918825986671046563883453843, 2.89246568810851457551008981085, 3.78104659619389634277410359957, 5.84701387775117610000187681430, 6.04079453419068894731151614697, 7.42207876331267788281322698772, 8.61098298149273721347111482480, 10.03531586646166499332589871652, 10.923285534516306831855993810836, 11.77568364708693107014263154750, 12.90670799194643304142794265892, 13.61440163086560008164308810429, 14.33963427626420656211455630211, 15.14415735824198399800427987582, 16.563788248044619883853391111788, 18.10828866006626300751000263189, 18.6085026782604152265528940958, 19.23820923740982535270069995897, 20.43391661424980644136212746692, 21.1798025884910031099766949667, 22.19744526201391392290314965520, 23.13658692204911005720150552731, 23.63929805943255745662261814772, 24.80037648343612454465817984008