L(s) = 1 | + (−0.669 − 0.743i)2-s + (0.309 − 0.951i)3-s + (−0.104 + 0.994i)4-s + (0.913 − 0.406i)5-s + (−0.913 + 0.406i)6-s + (0.978 + 0.207i)7-s + (0.809 − 0.587i)8-s + (−0.809 − 0.587i)9-s + (−0.913 − 0.406i)10-s − 11-s + (0.913 + 0.406i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.104 − 0.994i)15-s + (−0.978 − 0.207i)16-s + (0.104 − 0.994i)17-s + ⋯ |
L(s) = 1 | + (−0.669 − 0.743i)2-s + (0.309 − 0.951i)3-s + (−0.104 + 0.994i)4-s + (0.913 − 0.406i)5-s + (−0.913 + 0.406i)6-s + (0.978 + 0.207i)7-s + (0.809 − 0.587i)8-s + (−0.809 − 0.587i)9-s + (−0.913 − 0.406i)10-s − 11-s + (0.913 + 0.406i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.104 − 0.994i)15-s + (−0.978 − 0.207i)16-s + (0.104 − 0.994i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5451995669 - 0.6029503673i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5451995669 - 0.6029503673i\) |
\(L(1)\) |
\(\approx\) |
\(0.7482076253 - 0.5125493758i\) |
\(L(1)\) |
\(\approx\) |
\(0.7482076253 - 0.5125493758i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 61 | \( 1 \) |
good | 2 | \( 1 + (-0.669 - 0.743i)T \) |
| 3 | \( 1 + (0.309 - 0.951i)T \) |
| 5 | \( 1 + (0.913 - 0.406i)T \) |
| 7 | \( 1 + (0.978 + 0.207i)T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
| 17 | \( 1 + (0.104 - 0.994i)T \) |
| 19 | \( 1 + (-0.978 + 0.207i)T \) |
| 23 | \( 1 + (0.809 + 0.587i)T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (-0.669 + 0.743i)T \) |
| 37 | \( 1 + (-0.309 - 0.951i)T \) |
| 41 | \( 1 + (0.309 + 0.951i)T \) |
| 43 | \( 1 + (0.104 + 0.994i)T \) |
| 47 | \( 1 + (-0.5 - 0.866i)T \) |
| 53 | \( 1 + (0.809 - 0.587i)T \) |
| 59 | \( 1 + (-0.669 - 0.743i)T \) |
| 67 | \( 1 + (-0.913 + 0.406i)T \) |
| 71 | \( 1 + (-0.913 - 0.406i)T \) |
| 73 | \( 1 + (0.913 + 0.406i)T \) |
| 79 | \( 1 + (0.104 + 0.994i)T \) |
| 83 | \( 1 + (0.669 + 0.743i)T \) |
| 89 | \( 1 + (-0.309 + 0.951i)T \) |
| 97 | \( 1 + (0.669 - 0.743i)T \) |
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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
−32.97173788366851420674289753954, −32.24052556274513396968413255226, −30.75182386510041113868484189507, −29.21552982728841446530521064660, −28.03971999555484298383730794102, −27.10226935620048051728886110948, −26.15426183319436634450007248788, −25.37943137306110139816620996398, −24.13294992230035122106112836189, −22.74258942724319014940113049069, −21.38953428628502924084576305136, −20.46788586443802844127460981657, −18.94338796589199262894495814297, −17.57599841703519920694471159388, −16.955848642418354184799554112969, −15.26332479602001043775246470935, −14.73697617323359658269645799654, −13.45010175300192820063184750925, −10.68666107246087516877206298282, −10.346620337649597958005143151644, −8.82364257793633560234347341685, −7.72561211785368057714718512107, −5.84588125057169941271758671134, −4.76884256956956601295569032986, −2.326456695812353284848099762628,
1.56252436725873366610397198561, 2.59614927947992060354621861199, 5.04903541065735815784456018084, 7.07545766295571238266962713767, 8.35342973849612087331020206411, 9.35089890341811762034608095111, 10.96296922982038716909987574191, 12.24974991982701756554472967589, 13.25817048943825931911324749619, 14.390613924944570226968774686, 16.60553932829123426668508292799, 17.79753982268963355204983134855, 18.346803864854079230342968980749, 19.63855600849365428574093026289, 20.91173832213128542078382463970, 21.4681193701626490752760340177, 23.41274889412016046662951971737, 24.70356907184262223936724633660, 25.52101811005130630704624925282, 26.68351748260581963709635995783, 28.07048894504748056838273554363, 29.175386406127161954306775422712, 29.664593285799345352330127954462, 31.07967531864408093246346353549, 31.63032012285317097856245976216