L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s − 11-s + 12-s − 4·13-s + 16-s + 17-s − 18-s + 3·19-s + 22-s − 23-s − 24-s − 5·25-s + 4·26-s + 27-s − 29-s − 6·31-s − 32-s − 33-s − 34-s + 36-s − 3·37-s − 3·38-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s − 1.10·13-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.688·19-s + 0.213·22-s − 0.208·23-s − 0.204·24-s − 25-s + 0.784·26-s + 0.192·27-s − 0.185·29-s − 1.07·31-s − 0.176·32-s − 0.174·33-s − 0.171·34-s + 1/6·36-s − 0.493·37-s − 0.486·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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
−8.222980016689613755543054692732, −7.52924972171521652284941546776, −7.24751022640105054259238784813, −6.07792194424738019263601409287, −5.31591807918446499075250553141, −4.30091593132902903437897162799, −3.28365448396257876091801857917, −2.46145127550352097266597092633, −1.53559995770413638609622426592, 0,
1.53559995770413638609622426592, 2.46145127550352097266597092633, 3.28365448396257876091801857917, 4.30091593132902903437897162799, 5.31591807918446499075250553141, 6.07792194424738019263601409287, 7.24751022640105054259238784813, 7.52924972171521652284941546776, 8.222980016689613755543054692732