L(s) = 1 | + (0.990 − 0.139i)3-s + (0.0348 + 0.999i)4-s + (0.454 + 1.82i)5-s + (0.961 − 0.275i)9-s + (0.173 + 0.984i)12-s + (0.704 + 1.74i)15-s + (−0.997 + 0.0697i)16-s + (−1.80 + 0.518i)20-s + (0.939 − 0.342i)23-s + (−2.23 + 1.18i)25-s + (0.913 − 0.406i)27-s + (0.194 − 0.287i)31-s + (0.309 + 0.951i)36-s + (−1.25 − 1.39i)37-s + (0.939 + 1.62i)45-s + ⋯ |
L(s) = 1 | + (0.990 − 0.139i)3-s + (0.0348 + 0.999i)4-s + (0.454 + 1.82i)5-s + (0.961 − 0.275i)9-s + (0.173 + 0.984i)12-s + (0.704 + 1.74i)15-s + (−0.997 + 0.0697i)16-s + (−1.80 + 0.518i)20-s + (0.939 − 0.342i)23-s + (−2.23 + 1.18i)25-s + (0.913 − 0.406i)27-s + (0.194 − 0.287i)31-s + (0.309 + 0.951i)36-s + (−1.25 − 1.39i)37-s + (0.939 + 1.62i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0692 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0692 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.971977308\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.971977308\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.990 + 0.139i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.0348 - 0.999i)T^{2} \) |
| 5 | \( 1 + (-0.454 - 1.82i)T + (-0.882 + 0.469i)T^{2} \) |
| 7 | \( 1 + (0.241 + 0.970i)T^{2} \) |
| 13 | \( 1 + (0.615 - 0.788i)T^{2} \) |
| 17 | \( 1 + (-0.669 + 0.743i)T^{2} \) |
| 19 | \( 1 + (0.104 + 0.994i)T^{2} \) |
| 23 | \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 29 | \( 1 + (-0.961 - 0.275i)T^{2} \) |
| 31 | \( 1 + (-0.194 + 0.287i)T + (-0.374 - 0.927i)T^{2} \) |
| 37 | \( 1 + (1.25 + 1.39i)T + (-0.104 + 0.994i)T^{2} \) |
| 41 | \( 1 + (-0.961 + 0.275i)T^{2} \) |
| 43 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.0534 + 1.53i)T + (-0.997 - 0.0697i)T^{2} \) |
| 53 | \( 1 + (0.280 - 0.204i)T + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.294 + 0.184i)T + (0.438 - 0.898i)T^{2} \) |
| 61 | \( 1 + (0.374 - 0.927i)T^{2} \) |
| 67 | \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \) |
| 71 | \( 1 + (-1.39 + 0.623i)T + (0.669 - 0.743i)T^{2} \) |
| 73 | \( 1 + (-0.913 + 0.406i)T^{2} \) |
| 79 | \( 1 + (-0.0348 - 0.999i)T^{2} \) |
| 83 | \( 1 + (0.615 + 0.788i)T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.370 - 1.48i)T + (-0.882 - 0.469i)T^{2} \) |
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
−8.983600044546104880292726361349, −8.156061109644642054599042736825, −7.48308863390744822080522128754, −6.87655427335687735965292533521, −6.50744607697839932761091587471, −5.18266890669974966807444216042, −3.81445656970418279318343639448, −3.47060251444047789592889008289, −2.57763971443856897560263582003, −2.07790270737296179375248539311,
1.15819903551258542318064340152, 1.71175294769462543877296490872, 2.89283988339430504202352128297, 4.18690823040190943129771773824, 4.84574645871670612233668669941, 5.34716969042629869594216899870, 6.28526761160269782514050306832, 7.21717864403634309094735058088, 8.254269113247129443673693896948, 8.678063200730594640412964938621