L(s) = 1 | + 1.73i·3-s − 3.46·7-s − 2.99·9-s − 3.46·11-s + 4i·13-s + 6·17-s − 3.46i·19-s − 5.99i·21-s − 3.46i·23-s − 5.19i·27-s − 6i·29-s − 3.46i·31-s − 5.99i·33-s + 4i·37-s − 6.92·39-s + ⋯ |
L(s) = 1 | + 0.999i·3-s − 1.30·7-s − 0.999·9-s − 1.04·11-s + 1.10i·13-s + 1.45·17-s − 0.794i·19-s − 1.30i·21-s − 0.722i·23-s − 0.999i·27-s − 1.11i·29-s − 0.622i·31-s − 1.04i·33-s + 0.657i·37-s − 1.10·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0599 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0599 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3227412984\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3227412984\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 3.46iT - 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 + 12iT - 41T^{2} \) |
| 43 | \( 1 + 6.92T + 43T^{2} \) |
| 47 | \( 1 + 3.46iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 6.92T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - 10.3iT - 79T^{2} \) |
| 83 | \( 1 - 10.3iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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
−9.751097024153991486737584027478, −8.923950932926339665345100251167, −8.040859998437126273922459313723, −6.96153231702213991202372303889, −6.07309799562132128577814309949, −5.25520674165005389280692451914, −4.26529821317504420513894404820, −3.33557797184952633163442298521, −2.49288223119239946990374059845, −0.13913519217452760597168849125,
1.35616875253277572459387037971, 3.02466625771428007346740896713, 3.26070601934109285016083911164, 5.21391367531390661488275909399, 5.82147497366625092112464639473, 6.63064684602502409705402700515, 7.69468976842032500805903563471, 7.960274382343062594505721771394, 9.132984196246090727179682120483, 10.06078838463577685714102350712