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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.322693065 - 0.1824834269i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.322693065 - 0.1824834269i\) |
\(L(1)\) |
\(\approx\) |
\(1.430792096 - 0.1819230454i\) |
\(L(1)\) |
\(\approx\) |
\(1.430792096 - 0.1819230454i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
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 | \( 1 + (0.669 + 0.743i)T \) |
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
−32.391863970190405267605841522390, −31.80939837080270424711974087823, −30.29468282001178223599548466210, −29.66197689867740134189627943231, −28.48397660318353154033108655752, −26.35873426289461623461953219660, −25.64210335802645789767714211511, −24.706543763502428143141375489603, −23.88003218005248273061038739677, −22.59484519971203850650848808066, −21.55351117218188205353437110724, −20.10842488793530827540250784936, −18.83985556603394817949100825782, −17.213017551575537329373541764294, −16.769992976853064579755938221382, −14.85022049804415971928536227216, −13.81940249998491266663295594695, −12.95558410424600808597569449811, −12.01168181285223177270053498750, −9.53559184826933572078387697777, −8.33931292431334942744877995620, −6.61228097042314439867375709100, −6.15301111235568168505350274966, −4.057495415858771086653725497536, −2.211549375892348612868539608438,
2.46273271448130016417989021119, 3.57859383457997580860638934962, 5.208702132746276048466085223829, 6.4405469027192969994555300431, 9.23256734770494618073151566850, 9.877997229858242916979110674325, 11.023725604094676868604548366607, 12.62269269401520193021931964159, 13.90725567728124763861618227205, 14.78558925994019318559678847361, 16.03222548894384626652312981695, 17.62838155941287909163139258147, 19.27253866811969751900863281074, 20.13138003904431532991296461108, 21.39714796074035587166258120270, 22.2191618372792069612133848213, 22.795489388035498777773342719372, 24.84933249659629794030578331561, 25.709559417117719228408597599583, 27.16515341742261067728078370957, 28.16219044407554605775243812062, 29.37724076545146552730253568809, 30.11735352705525191492315191749, 31.69210462522182605107341766617, 32.35629395035699339213198570380