L(s) = 1 | + (0.5 − 0.866i)3-s + 5-s + (0.5 + 0.866i)7-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + 21-s + (−0.5 + 0.866i)23-s + 25-s − 27-s + (0.5 − 0.866i)29-s − 31-s + (0.5 + 0.866i)33-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)3-s + 5-s + (0.5 + 0.866i)7-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + 21-s + (−0.5 + 0.866i)23-s + 25-s − 27-s + (0.5 − 0.866i)29-s − 31-s + (0.5 + 0.866i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.288691360 - 0.3541753007i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.288691360 - 0.3541753007i\) |
\(L(1)\) |
\(\approx\) |
\(1.291249177 - 0.2501954824i\) |
\(L(1)\) |
\(\approx\) |
\(1.291249177 - 0.2501954824i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (-0.5 + 0.866i)T \) |
| 41 | \( 1 + (0.5 - 0.866i)T \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + (-0.5 - 0.866i)T \) |
| 61 | \( 1 + (0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.5 + 0.866i)T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 97 | \( 1 + (0.5 + 0.866i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−29.81586820532432805112310685953, −28.84933101595610237926930517079, −27.63918545054613781160558403847, −26.605641534689777448654442904577, −25.99287881506459123733209241462, −24.831005014865795851054914724030, −23.7253388291041530634555556476, −22.281344209761556947616247352111, −21.35928941018932146496213375818, −20.70738441400063213283540007900, −19.605166085965235388409903353459, −18.17105865990686052049339728585, −16.971096852278158221962301372017, −16.207706636072382336397519448637, −14.64128957029327589816662713904, −14.018985809867092670334959000001, −12.91673434139402087338801804816, −10.79139040032724931020498032332, −10.4219472408708471009070416485, −8.996625810336510938766628070987, −7.97922932543144328821640871969, −6.16580115473281837296560030197, −4.87694529924387650397855641807, −3.56738605753523571469733953735, −1.97907815088591352776868015566,
1.83671570534581056424627007949, 2.66582896025773568177154364920, 4.96450678752792912527998585715, 6.22152181351879384100683978153, 7.4542251285324526000363880529, 8.77539317905238582073472233273, 9.69639223867933151650055163623, 11.40904551344656881498173382550, 12.62200893120899447147747676213, 13.511268100249686205616823139608, 14.56031988931439039735250853248, 15.61529440100672636384711811087, 17.607783996096574856136222338957, 17.8861339601659023626371962781, 19.03835222774413081820218015181, 20.34099107269857651734273265653, 21.20006426423752059882264236121, 22.35623470312590826791790717213, 23.730027452685174125172927551574, 24.68692144833118609006413399824, 25.454370770400769994206342534105, 26.183952813776923229144697795545, 27.827185010921866608608998780351, 28.82687424016998694947203978869, 29.660863332731324942413666526689