L(s) = 1 | − 2i·7-s + 4·11-s − 2i·13-s − 5i·17-s + 5·19-s + i·23-s + 2·29-s + 7·31-s + 6i·37-s + 4i·43-s − 4i·47-s + 3·49-s + 9i·53-s − 14·59-s − 11·61-s + ⋯ |
L(s) = 1 | − 0.755i·7-s + 1.20·11-s − 0.554i·13-s − 1.21i·17-s + 1.14·19-s + 0.208i·23-s + 0.371·29-s + 1.25·31-s + 0.986i·37-s + 0.609i·43-s − 0.583i·47-s + 0.428·49-s + 1.23i·53-s − 1.82·59-s − 1.40·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.201448783\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.201448783\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 5iT - 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 - iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 9iT - 53T^{2} \) |
| 59 | \( 1 + 14T + 59T^{2} \) |
| 61 | \( 1 + 11T + 61T^{2} \) |
| 67 | \( 1 + 14iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 12iT - 73T^{2} \) |
| 79 | \( 1 - 3T + 79T^{2} \) |
| 83 | \( 1 + iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 16iT - 97T^{2} \) |
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
−7.76350846739432601511761921944, −7.50967502341786347062760270551, −6.57721799896623568344948336535, −6.06788941703415977485679885864, −4.93462674203343226504718087974, −4.50001122267334841834769597219, −3.42006851196004298677118705664, −2.90374307619972749451413630349, −1.46371370780795108627137448942, −0.69059870266953878768696197914,
1.10339792779453947612749867421, 1.99195114414360831362003479552, 3.00032659809064789519806859838, 3.88678061193313912068876285760, 4.55130856256476425388310426825, 5.54095793334687887579363663852, 6.16725778531207470312957506781, 6.76036459566797312965647007126, 7.59832217774317591493071614243, 8.433019509621554621510337574800