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
−10.06078838463577685714102350712, −9.132984196246090727179682120483, −7.960274382343062594505721771394, −7.69468976842032500805903563471, −6.63064684602502409705402700515, −5.82147497366625092112464639473, −5.21391367531390661488275909399, −3.26070601934109285016083911164, −3.02466625771428007346740896713, −1.35616875253277572459387037971,
0.13913519217452760597168849125, 2.49288223119239946990374059845, 3.33557797184952633163442298521, 4.26529821317504420513894404820, 5.25520674165005389280692451914, 6.07309799562132128577814309949, 6.96153231702213991202372303889, 8.040859998437126273922459313723, 8.923950932926339665345100251167, 9.751097024153991486737584027478