L(s) = 1 | + 5-s − 4·7-s − 11-s + 4·13-s + 4·19-s + 6·23-s + 25-s − 6·29-s + 8·31-s − 4·35-s − 2·37-s − 6·41-s − 8·43-s − 6·47-s + 9·49-s − 6·53-s − 55-s − 12·59-s − 2·61-s + 4·65-s + 10·67-s + 12·71-s − 16·73-s + 4·77-s + 8·79-s − 6·89-s − 16·91-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.51·7-s − 0.301·11-s + 1.10·13-s + 0.917·19-s + 1.25·23-s + 1/5·25-s − 1.11·29-s + 1.43·31-s − 0.676·35-s − 0.328·37-s − 0.937·41-s − 1.21·43-s − 0.875·47-s + 9/7·49-s − 0.824·53-s − 0.134·55-s − 1.56·59-s − 0.256·61-s + 0.496·65-s + 1.22·67-s + 1.42·71-s − 1.87·73-s + 0.455·77-s + 0.900·79-s − 0.635·89-s − 1.67·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−15.39316702015709, −14.97447768714718, −14.07276351975540, −13.61584873062668, −13.26703480120546, −12.84210185993378, −12.28712968471581, −11.54328946420625, −11.10690378029400, −10.38609929946429, −9.959807498755737, −9.420051622861581, −9.013664404600185, −8.326849008146527, −7.703539175578376, −6.864068210875798, −6.548542835018125, −6.019938130190735, −5.319936183145588, −4.794148195713737, −3.771031466570881, −3.206210872458957, −2.906216023726088, −1.789546095593508, −1.018706115886145, 0,
1.018706115886145, 1.789546095593508, 2.906216023726088, 3.206210872458957, 3.771031466570881, 4.794148195713737, 5.319936183145588, 6.019938130190735, 6.548542835018125, 6.864068210875798, 7.703539175578376, 8.326849008146527, 9.013664404600185, 9.420051622861581, 9.959807498755737, 10.38609929946429, 11.10690378029400, 11.54328946420625, 12.28712968471581, 12.84210185993378, 13.26703480120546, 13.61584873062668, 14.07276351975540, 14.97447768714718, 15.39316702015709