L(s) = 1 | + 3-s − 2·7-s + 9-s + 11-s − 4·13-s + 6·17-s + 4·19-s − 2·21-s − 6·23-s + 27-s − 6·29-s + 8·31-s + 33-s − 10·37-s − 4·39-s + 6·41-s + 8·43-s + 6·47-s − 3·49-s + 6·51-s + 4·57-s − 8·61-s − 2·63-s − 4·67-s − 6·69-s + 6·71-s − 2·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s + 1.45·17-s + 0.917·19-s − 0.436·21-s − 1.25·23-s + 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.174·33-s − 1.64·37-s − 0.640·39-s + 0.937·41-s + 1.21·43-s + 0.875·47-s − 3/7·49-s + 0.840·51-s + 0.529·57-s − 1.02·61-s − 0.251·63-s − 0.488·67-s − 0.722·69-s + 0.712·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 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 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.56629349852309, −14.18307400881615, −13.82283320435255, −13.30757845687408, −12.44757861473134, −12.23581573758743, −11.97456043611663, −11.07613195464807, −10.41627494438286, −9.936394893569758, −9.500492888404769, −9.266751579225610, −8.365322726979085, −7.842682180227686, −7.421909703790472, −6.934145791909805, −6.171402841751341, −5.630781584530007, −5.108530194943379, −4.225142174114762, −3.774360732104261, −3.058481317637003, −2.645267125971276, −1.791269508041175, −0.9996751412569233, 0,
0.9996751412569233, 1.791269508041175, 2.645267125971276, 3.058481317637003, 3.774360732104261, 4.225142174114762, 5.108530194943379, 5.630781584530007, 6.171402841751341, 6.934145791909805, 7.421909703790472, 7.842682180227686, 8.365322726979085, 9.266751579225610, 9.500492888404769, 9.936394893569758, 10.41627494438286, 11.07613195464807, 11.97456043611663, 12.23581573758743, 12.44757861473134, 13.30757845687408, 13.82283320435255, 14.18307400881615, 14.56629349852309