| L(s) = 1 | + 1.73i·5-s + 3.46·7-s + 1.73i·13-s + 3·17-s + 2i·19-s + 6.92·23-s + 2.00·25-s − 8.66i·29-s + 3.46·31-s + 5.99i·35-s + 8.66i·37-s − 6·41-s + 4i·43-s − 3.46·47-s + 4.99·49-s + ⋯ |
| L(s) = 1 | + 0.774i·5-s + 1.30·7-s + 0.480i·13-s + 0.727·17-s + 0.458i·19-s + 1.44·23-s + 0.400·25-s − 1.60i·29-s + 0.622·31-s + 1.01i·35-s + 1.42i·37-s − 0.937·41-s + 0.609i·43-s − 0.505·47-s + 0.714·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.576655086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.576655086\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1.73iT - 5T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 1.73iT - 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 + 8.66iT - 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 - 8.66iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 + 6.92iT - 53T^{2} \) |
| 59 | \( 1 + 12iT - 59T^{2} \) |
| 61 | \( 1 + 5.19iT - 61T^{2} \) |
| 67 | \( 1 - 10iT - 67T^{2} \) |
| 71 | \( 1 - 3.46T + 71T^{2} \) |
| 73 | \( 1 - 5T + 73T^{2} \) |
| 79 | \( 1 - 6.92T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 15T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
| 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
−8.013969622465683060306021235900, −7.84791081984556658275960808791, −6.72976838295901895585767328097, −6.38594794890546208987485827427, −5.15143905217675368645495671480, −4.83164939687497247460349988886, −3.76361921173509362645592349059, −2.94574649806288194348277673802, −2.00747825093852426817748101850, −1.06058240424314094067181877744,
0.857264959573225923100842040113, 1.53504387517983267469670157003, 2.71477994345080444098942382861, 3.63670012625143761838429755792, 4.77697112598649418321675666454, 5.01953746101085434385884086285, 5.69912171976496974490663888433, 6.85610400602415655757070733705, 7.48311870287596769854653816049, 8.173809851131573404340147816849
