L(s) = 1 | − 3-s + 0.922i·5-s + (2.06 − 1.65i)7-s + 9-s − 1.61i·11-s + 2.82i·13-s − 0.922i·15-s + 0.922i·17-s + 0.540·19-s + (−2.06 + 1.65i)21-s − 7.05i·23-s + 4.14·25-s − 27-s + 1.01·29-s + 8.67·31-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.412i·5-s + (0.781 − 0.624i)7-s + 0.333·9-s − 0.487i·11-s + 0.784i·13-s − 0.238i·15-s + 0.223i·17-s + 0.123·19-s + (−0.450 + 0.360i)21-s − 1.47i·23-s + 0.829·25-s − 0.192·27-s + 0.188·29-s + 1.55·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.37581 - 0.0763885i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.37581 - 0.0763885i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + (-2.06 + 1.65i)T \) |
good | 5 | \( 1 - 0.922iT - 5T^{2} \) |
| 11 | \( 1 + 1.61iT - 11T^{2} \) |
| 13 | \( 1 - 2.82iT - 13T^{2} \) |
| 17 | \( 1 - 0.922iT - 17T^{2} \) |
| 19 | \( 1 - 0.540T + 19T^{2} \) |
| 23 | \( 1 + 7.05iT - 23T^{2} \) |
| 29 | \( 1 - 1.01T + 29T^{2} \) |
| 31 | \( 1 - 8.67T + 31T^{2} \) |
| 37 | \( 1 - 8.19T + 37T^{2} \) |
| 41 | \( 1 - 6.57iT - 41T^{2} \) |
| 43 | \( 1 - 4.96iT - 43T^{2} \) |
| 47 | \( 1 - 2.60T + 47T^{2} \) |
| 53 | \( 1 - 5.59T + 53T^{2} \) |
| 59 | \( 1 - 1.65T + 59T^{2} \) |
| 61 | \( 1 + 7.01iT - 61T^{2} \) |
| 67 | \( 1 - 6.80iT - 67T^{2} \) |
| 71 | \( 1 + 5.21iT - 71T^{2} \) |
| 73 | \( 1 + 7.50iT - 73T^{2} \) |
| 79 | \( 1 + 7.11iT - 79T^{2} \) |
| 83 | \( 1 + 17.9T + 83T^{2} \) |
| 89 | \( 1 - 17.9iT - 89T^{2} \) |
| 97 | \( 1 + 4.18iT - 97T^{2} \) |
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
−10.63718805955167987703286832360, −9.857994986560418059227999909767, −8.648972287972147992963750482344, −7.86522456721895799892205411210, −6.78806197549801688663838293581, −6.19858511858803703957232927877, −4.84982693620670719116515227344, −4.19377472299050687178511805027, −2.67802252048264656560045916424, −1.06123788719236089520345747959,
1.16556609990206262074365838610, 2.64542618365220786862307016810, 4.22902707303103262258035318695, 5.18911218524854693542817094170, 5.76121692051386185907343282082, 7.04017446435037993483323890131, 7.919891378173166098451520783719, 8.768970004986895051829494395465, 9.716017398920146445468816061661, 10.55735835879705620456582384503