L(s) = 1 | + (−0.432 − 2.19i)5-s − i·7-s + 0.626·11-s + 5.49i·13-s + 0.896i·17-s − 6.38·19-s + 3.72i·23-s + (−4.62 + 1.89i)25-s − 7.87·29-s + 7.52·31-s + (−2.19 + 0.432i)35-s − 6i·37-s − 7.72·41-s + 1.72i·43-s + 5.87i·47-s + ⋯ |
L(s) = 1 | + (−0.193 − 0.981i)5-s − 0.377i·7-s + 0.188·11-s + 1.52i·13-s + 0.217i·17-s − 1.46·19-s + 0.777i·23-s + (−0.925 + 0.379i)25-s − 1.46·29-s + 1.35·31-s + (−0.370 + 0.0730i)35-s − 0.986i·37-s − 1.20·41-s + 0.263i·43-s + 0.857i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.193 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.193 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7267924714\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7267924714\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.432 + 2.19i)T \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 - 0.626T + 11T^{2} \) |
| 13 | \( 1 - 5.49iT - 13T^{2} \) |
| 17 | \( 1 - 0.896iT - 17T^{2} \) |
| 19 | \( 1 + 6.38T + 19T^{2} \) |
| 23 | \( 1 - 3.72iT - 23T^{2} \) |
| 29 | \( 1 + 7.87T + 29T^{2} \) |
| 31 | \( 1 - 7.52T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 7.72T + 41T^{2} \) |
| 43 | \( 1 - 1.72iT - 43T^{2} \) |
| 47 | \( 1 - 5.87iT - 47T^{2} \) |
| 53 | \( 1 - 6.77iT - 53T^{2} \) |
| 59 | \( 1 + 0.593T + 59T^{2} \) |
| 61 | \( 1 - 7.13T + 61T^{2} \) |
| 67 | \( 1 - 5.79iT - 67T^{2} \) |
| 71 | \( 1 + 5.52T + 71T^{2} \) |
| 73 | \( 1 + 3.72iT - 73T^{2} \) |
| 79 | \( 1 - 5.67T + 79T^{2} \) |
| 83 | \( 1 - 17.4iT - 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 - 10.1iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.142441189410484905166653263170, −8.432950799284850389293339365280, −7.68059516023530577485363769882, −6.80600990180376979819135352613, −6.08410123739614494186195876542, −5.08886334508736141219828756757, −4.23427601888781306023596922337, −3.81465500942225348979093730084, −2.20520732190493527744107221225, −1.31537123417937857453737081584,
0.23966721023130268378553724670, 2.03132308992885941311253924239, 2.91335398132730255943807104845, 3.67839226278887463688795582420, 4.74478143579938009519324709833, 5.69779207375278988211679766115, 6.43951333555533942099127685623, 7.06279155148762369615571652429, 8.123081565105557549965718406296, 8.412562117943245801723394009744