L(s) = 1 | + i·3-s + (−1.19 − 1.88i)5-s + 2.82i·7-s − 9-s + 0.393·11-s − i·13-s + (1.88 − 1.19i)15-s − 6.21i·17-s + 7.34·19-s − 2.82·21-s + 1.44i·23-s + (−2.13 + 4.52i)25-s − i·27-s + 0.828·29-s + 1.65·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + (−0.535 − 0.844i)5-s + 1.06i·7-s − 0.333·9-s + 0.118·11-s − 0.277i·13-s + (0.487 − 0.308i)15-s − 1.50i·17-s + 1.68·19-s − 0.617·21-s + 0.301i·23-s + (−0.427 + 0.904i)25-s − 0.192i·27-s + 0.153·29-s + 0.297·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.514524268\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.514524268\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (1.19 + 1.88i)T \) |
| 13 | \( 1 + iT \) |
good | 7 | \( 1 - 2.82iT - 7T^{2} \) |
| 11 | \( 1 - 0.393T + 11T^{2} \) |
| 17 | \( 1 + 6.21iT - 17T^{2} \) |
| 19 | \( 1 - 7.34T + 19T^{2} \) |
| 23 | \( 1 - 1.44iT - 23T^{2} \) |
| 29 | \( 1 - 0.828T + 29T^{2} \) |
| 31 | \( 1 - 1.65T + 31T^{2} \) |
| 37 | \( 1 - 6.78iT - 37T^{2} \) |
| 41 | \( 1 - 3.77T + 41T^{2} \) |
| 43 | \( 1 - 2.51iT - 43T^{2} \) |
| 47 | \( 1 - 3.00iT - 47T^{2} \) |
| 53 | \( 1 - 9.55iT - 53T^{2} \) |
| 59 | \( 1 + 0.393T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 - 15.5iT - 67T^{2} \) |
| 71 | \( 1 - 6.56T + 71T^{2} \) |
| 73 | \( 1 + 13.6iT - 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 + 16.4iT - 83T^{2} \) |
| 89 | \( 1 - 9.67T + 89T^{2} \) |
| 97 | \( 1 - 0.0418iT - 97T^{2} \) |
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
−9.335085274910427305131338583986, −8.943351934776913671598995192116, −7.989533383972435431094855761447, −7.31173322925030056336138720417, −5.99760270435034360899328272121, −5.19067454313575150573731981519, −4.72980084130521870132613349675, −3.47966914888879384451106497508, −2.64434246040363223438481490201, −0.959980320387941869754719220823,
0.819295712751192257731995677036, 2.18907882736173836328995768789, 3.49555492351764625154428319522, 3.99615017152054628247184647159, 5.32340419944529965841065712418, 6.42127077039447582069007626487, 6.96726801603892054113298035055, 7.71245999452670598898556053826, 8.255385340887474081418610165750, 9.441121417004119848500425157470