| L(s) = 1 | + 5-s − 3·7-s + 3·13-s + 4·17-s + 19-s + 25-s − 3·29-s + 7·31-s − 3·35-s + 8·37-s + 7·41-s − 43-s + 6·47-s + 2·49-s − 6·53-s − 4·59-s − 7·61-s + 3·65-s − 5·67-s − 2·71-s − 73-s + 9·79-s + 8·83-s + 4·85-s − 4·89-s − 9·91-s + 95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.13·7-s + 0.832·13-s + 0.970·17-s + 0.229·19-s + 1/5·25-s − 0.557·29-s + 1.25·31-s − 0.507·35-s + 1.31·37-s + 1.09·41-s − 0.152·43-s + 0.875·47-s + 2/7·49-s − 0.824·53-s − 0.520·59-s − 0.896·61-s + 0.372·65-s − 0.610·67-s − 0.237·71-s − 0.117·73-s + 1.01·79-s + 0.878·83-s + 0.433·85-s − 0.423·89-s − 0.943·91-s + 0.102·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−13.55550529738481, −13.40121188375379, −12.90100339269431, −12.23964629506267, −12.08981496781838, −11.29189041580784, −10.79987808859154, −10.39509267540288, −9.775650684609626, −9.398571909316305, −9.142713895008682, −8.330973569120321, −7.848416670721553, −7.409281602299443, −6.640513327425979, −6.218147603045278, −5.928472508033323, −5.332614487831200, −4.620654143683462, −3.985796545826069, −3.457689002399995, −2.855307353613767, −2.453691266794502, −1.389537868281514, −0.9782620037927084, 0,
0.9782620037927084, 1.389537868281514, 2.453691266794502, 2.855307353613767, 3.457689002399995, 3.985796545826069, 4.620654143683462, 5.332614487831200, 5.928472508033323, 6.218147603045278, 6.640513327425979, 7.409281602299443, 7.848416670721553, 8.330973569120321, 9.142713895008682, 9.398571909316305, 9.775650684609626, 10.39509267540288, 10.79987808859154, 11.29189041580784, 12.08981496781838, 12.23964629506267, 12.90100339269431, 13.40121188375379, 13.55550529738481
