L(s) = 1 | − 3-s − 2·7-s − 2·9-s + 5·11-s − 5·17-s + 5·19-s + 2·21-s + 6·23-s + 5·27-s − 4·29-s − 10·31-s − 5·33-s − 10·37-s + 5·41-s − 4·43-s − 8·47-s − 3·49-s + 5·51-s − 10·53-s − 5·57-s + 10·61-s + 4·63-s − 3·67-s − 6·69-s + 5·73-s − 10·77-s − 10·79-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.755·7-s − 2/3·9-s + 1.50·11-s − 1.21·17-s + 1.14·19-s + 0.436·21-s + 1.25·23-s + 0.962·27-s − 0.742·29-s − 1.79·31-s − 0.870·33-s − 1.64·37-s + 0.780·41-s − 0.609·43-s − 1.16·47-s − 3/7·49-s + 0.700·51-s − 1.37·53-s − 0.662·57-s + 1.28·61-s + 0.503·63-s − 0.366·67-s − 0.722·69-s + 0.585·73-s − 1.13·77-s − 1.12·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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
−9.264342318718394197961750382476, −8.378729337077873331827570433737, −6.99159588501341969315252403426, −6.73749937778767246819441820947, −5.74787484984166059088095934275, −5.01042933129726602196570696286, −3.81174964693589815456776232954, −3.05242102020450397566462398303, −1.54134493970211049092330346485, 0,
1.54134493970211049092330346485, 3.05242102020450397566462398303, 3.81174964693589815456776232954, 5.01042933129726602196570696286, 5.74787484984166059088095934275, 6.73749937778767246819441820947, 6.99159588501341969315252403426, 8.378729337077873331827570433737, 9.264342318718394197961750382476