L(s) = 1 | + 1.41i·2-s − 2.00·4-s + 2.82i·5-s + 2·7-s − 2.82i·8-s − 4.00·10-s − 5.65i·11-s + 2.82i·14-s + 4.00·16-s − 5.65i·20-s + 8.00·22-s − 3.00·25-s − 4.00·28-s + 2.82i·29-s − 10·31-s + 5.65i·32-s + ⋯ |
L(s) = 1 | + 0.999i·2-s − 1.00·4-s + 1.26i·5-s + 0.755·7-s − 1.00i·8-s − 1.26·10-s − 1.70i·11-s + 0.755i·14-s + 1.00·16-s − 1.26i·20-s + 1.70·22-s − 0.600·25-s − 0.755·28-s + 0.525i·29-s − 1.79·31-s + 1.00i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.626059 + 0.626059i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.626059 + 0.626059i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 2.82iT - 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 14.1iT - 53T^{2} \) |
| 59 | \( 1 - 11.3iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 5.65iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−14.67775387191730282028398531410, −14.28909176397700135196412256125, −13.18532026363482569767025347565, −11.40237991051490769499946060660, −10.51537962851961349002635536516, −8.916367444552218392773030973974, −7.80946111775523970186167491896, −6.63376132110711059298421719352, −5.44701643907424513444627594829, −3.49994922448227040508054853507,
1.78250116892438199199236003766, 4.31228352047829502616019460801, 5.19534403732119005925496553562, 7.70320398882244853493446925815, 8.920752578114195302722929551822, 9.831838475826753763234022542131, 11.19071334099707376619912511674, 12.35718091867383862347936968396, 12.84210040565858777461327977235, 14.17434646576563573812079270815