L(s) = 1 | − 3-s − 7-s − 2·9-s − 3·11-s + 2·13-s + 3·17-s + 7·19-s + 21-s + 5·27-s − 6·29-s + 4·31-s + 3·33-s + 8·37-s − 2·39-s − 9·41-s − 8·43-s + 6·47-s + 49-s − 3·51-s − 12·53-s − 7·57-s − 12·59-s − 10·61-s + 2·63-s + 7·67-s − 6·71-s + 5·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s − 2/3·9-s − 0.904·11-s + 0.554·13-s + 0.727·17-s + 1.60·19-s + 0.218·21-s + 0.962·27-s − 1.11·29-s + 0.718·31-s + 0.522·33-s + 1.31·37-s − 0.320·39-s − 1.40·41-s − 1.21·43-s + 0.875·47-s + 1/7·49-s − 0.420·51-s − 1.64·53-s − 0.927·57-s − 1.56·59-s − 1.28·61-s + 0.251·63-s + 0.855·67-s − 0.712·71-s + 0.585·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 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
−8.261780224373543856297235747442, −7.76136885452014769391786338919, −6.84082843689233258487775944714, −5.92654096551589702125781476334, −5.48776820834083428423049797351, −4.68405477142379205905809833243, −3.37835476194466045277789097411, −2.85425374545583595819763274865, −1.33133249048010073515633278194, 0,
1.33133249048010073515633278194, 2.85425374545583595819763274865, 3.37835476194466045277789097411, 4.68405477142379205905809833243, 5.48776820834083428423049797351, 5.92654096551589702125781476334, 6.84082843689233258487775944714, 7.76136885452014769391786338919, 8.261780224373543856297235747442