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 | \( 1 + 10.1iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.412562117943245801723394009744, −8.123081565105557549965718406296, −7.06279155148762369615571652429, −6.43951333555533942099127685623, −5.69779207375278988211679766115, −4.74478143579938009519324709833, −3.67839226278887463688795582420, −2.91335398132730255943807104845, −2.03132308992885941311253924239, −0.23966721023130268378553724670,
1.31537123417937857453737081584, 2.20520732190493527744107221225, 3.81465500942225348979093730084, 4.23427601888781306023596922337, 5.08886334508736141219828756757, 6.08410123739614494186195876542, 6.80600990180376979819135352613, 7.68059516023530577485363769882, 8.432950799284850389293339365280, 9.142441189410484905166653263170