L(s) = 1 | + 3-s + 4i·7-s − 2·9-s − 3i·11-s − i·17-s − 7i·19-s + 4i·21-s − 4i·23-s − 5·27-s − 8i·29-s − 4·31-s − 3i·33-s + 4·37-s + 3·41-s − 8·43-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.51i·7-s − 0.666·9-s − 0.904i·11-s − 0.242i·17-s − 1.60i·19-s + 0.872i·21-s − 0.834i·23-s − 0.962·27-s − 1.48i·29-s − 0.718·31-s − 0.522i·33-s + 0.657·37-s + 0.468·41-s − 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.546495156\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.546495156\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + 3iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + iT - 17T^{2} \) |
| 19 | \( 1 + 7iT - 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 8iT - 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 12T + 53T^{2} \) |
| 59 | \( 1 - 8iT - 59T^{2} \) |
| 61 | \( 1 + 4iT - 61T^{2} \) |
| 67 | \( 1 - 9T + 67T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 + 11iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + T + 83T^{2} \) |
| 89 | \( 1 + 13T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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
−8.478682693682227360986449381633, −8.155670886314316423948009514420, −7.00776740855732643635488433377, −6.10803053616986082554035973017, −5.58289979669047444827529227913, −4.74946690227103099310791070625, −3.56858947763654977949171364857, −2.62794821220918021775694594016, −2.31919723744166946149789178290, −0.44637383891249922654298228084,
1.22902502393988523545545152192, 2.20137801989844879994723769684, 3.59438685322385097325814238541, 3.76460557451221756239824423097, 4.93208177667144126118351092043, 5.75169458892896113328667769082, 6.80412177784723650676077403747, 7.39402164892776952434994347599, 8.012570903099387420093175593332, 8.696438118881425273346399002281