L(s) = 1 | − 3·7-s − 2·11-s + 3·13-s + 6·17-s − 7·19-s + 6·23-s + 2·29-s − 5·31-s − 10·37-s − 12·41-s − 3·43-s − 10·47-s + 2·49-s + 6·59-s − 13·61-s − 7·67-s + 4·71-s + 6·73-s + 6·77-s − 8·79-s − 6·83-s − 16·89-s − 9·91-s + 7·97-s + 12·103-s − 16·107-s + 9·109-s + ⋯ |
L(s) = 1 | − 1.13·7-s − 0.603·11-s + 0.832·13-s + 1.45·17-s − 1.60·19-s + 1.25·23-s + 0.371·29-s − 0.898·31-s − 1.64·37-s − 1.87·41-s − 0.457·43-s − 1.45·47-s + 2/7·49-s + 0.781·59-s − 1.66·61-s − 0.855·67-s + 0.474·71-s + 0.702·73-s + 0.683·77-s − 0.900·79-s − 0.658·83-s − 1.69·89-s − 0.943·91-s + 0.710·97-s + 1.18·103-s − 1.54·107-s + 0.862·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 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
−8.772421839183462887904267277344, −8.279043826836005007614143835778, −7.14524835419643162911975279792, −6.53690929538739447537109821251, −5.70088645188412764957726143386, −4.85182738731759099554593417847, −3.55277063573082460622069589535, −3.08865327644648973893039079227, −1.63004456496430691453589862078, 0,
1.63004456496430691453589862078, 3.08865327644648973893039079227, 3.55277063573082460622069589535, 4.85182738731759099554593417847, 5.70088645188412764957726143386, 6.53690929538739447537109821251, 7.14524835419643162911975279792, 8.279043826836005007614143835778, 8.772421839183462887904267277344