L(s) = 1 | + 3i·5-s − 1.73i·7-s + 5.19·11-s + 2·13-s − 6i·17-s − 6.92i·19-s − 4·25-s + 6i·29-s − 5.19i·31-s + 5.19·35-s − 8·37-s + 10.3i·43-s + 10.3·47-s + 4·49-s − 9i·53-s + ⋯ |
L(s) = 1 | + 1.34i·5-s − 0.654i·7-s + 1.56·11-s + 0.554·13-s − 1.45i·17-s − 1.58i·19-s − 0.800·25-s + 1.11i·29-s − 0.933i·31-s + 0.878·35-s − 1.31·37-s + 1.58i·43-s + 1.51·47-s + 0.571·49-s − 1.23i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.934827741\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.934827741\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 3iT - 5T^{2} \) |
| 7 | \( 1 + 1.73iT - 7T^{2} \) |
| 11 | \( 1 - 5.19T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 6.92iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 + 5.19iT - 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 10.3iT - 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 9iT - 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 - 3.46iT - 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - T + 73T^{2} \) |
| 79 | \( 1 + 3.46iT - 79T^{2} \) |
| 83 | \( 1 + 5.19T + 83T^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 + 5T + 97T^{2} \) |
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
−9.349430018754690892627097515960, −8.681129711471335363924712308743, −7.37544013035595651079439416360, −6.90647291291446221822881990012, −6.48606728258172651779128529516, −5.23580117772196954355247479927, −4.13947901801013514964300761021, −3.37529558553623354544624888830, −2.45104155301302268322869647163, −0.919011476211340647445773075409,
1.16012666804914889793893556240, 1.93600067004104086248348938736, 3.77796534111514868149395069128, 4.09308215720697952450186769509, 5.43593328386143964436174316009, 5.90618445745239695545119098405, 6.80946026581393295350425941531, 8.070740179229920224203064774766, 8.693180799661226862809901034165, 9.005599075291961197632721012947