L(s) = 1 | − 2-s − 3-s − 4-s + 3·5-s + 6-s + 3·8-s + 9-s − 3·10-s + 3·11-s + 12-s + 3·13-s − 3·15-s − 16-s − 18-s + 8·19-s − 3·20-s − 3·22-s − 23-s − 3·24-s + 4·25-s − 3·26-s − 27-s + 29-s + 3·30-s + 7·31-s − 5·32-s − 3·33-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 1.34·5-s + 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.948·10-s + 0.904·11-s + 0.288·12-s + 0.832·13-s − 0.774·15-s − 1/4·16-s − 0.235·18-s + 1.83·19-s − 0.670·20-s − 0.639·22-s − 0.208·23-s − 0.612·24-s + 4/5·25-s − 0.588·26-s − 0.192·27-s + 0.185·29-s + 0.547·30-s + 1.25·31-s − 0.883·32-s − 0.522·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.328075112\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.328075112\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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
−9.276520723772856374909731788547, −8.629106599005923063947595537705, −7.67383403059463113074950694299, −6.72745029329513701643714206604, −5.99160036699561048994465346327, −5.28097963895318147961411148056, −4.41649055843207850390968347010, −3.26428207845127101346332288007, −1.68282741605517910914830386033, −0.997776941519556602195874588653,
0.997776941519556602195874588653, 1.68282741605517910914830386033, 3.26428207845127101346332288007, 4.41649055843207850390968347010, 5.28097963895318147961411148056, 5.99160036699561048994465346327, 6.72745029329513701643714206604, 7.67383403059463113074950694299, 8.629106599005923063947595537705, 9.276520723772856374909731788547