| L(s) = 1 | + 5-s − 2·7-s + 13-s − 4·17-s + 2·19-s + 7·23-s − 4·25-s − 2·29-s + 3·31-s − 2·35-s − 11·37-s + 10·41-s − 4·43-s − 4·47-s − 3·49-s − 2·53-s − 59-s + 2·61-s + 65-s + 67-s − 9·71-s + 16·73-s + 8·79-s − 4·85-s + 7·89-s − 2·91-s + 2·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s + 0.277·13-s − 0.970·17-s + 0.458·19-s + 1.45·23-s − 4/5·25-s − 0.371·29-s + 0.538·31-s − 0.338·35-s − 1.80·37-s + 1.56·41-s − 0.609·43-s − 0.583·47-s − 3/7·49-s − 0.274·53-s − 0.130·59-s + 0.256·61-s + 0.124·65-s + 0.122·67-s − 1.06·71-s + 1.87·73-s + 0.900·79-s − 0.433·85-s + 0.741·89-s − 0.209·91-s + 0.205·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 226512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 226512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−13.17229928775125, −12.89773917210681, −12.23759182303610, −11.84405902601713, −11.24092372526028, −10.82450843045894, −10.46370974138635, −9.790915856005430, −9.343015661540156, −9.205540734438780, −8.475605348638811, −8.082336206689725, −7.359064025777045, −6.942967720268963, −6.454935258051028, −6.125646129454446, −5.383400244029357, −5.073895001712893, −4.415765143102789, −3.770509601440700, −3.284524852629678, −2.770573339394819, −2.108840581088215, −1.541376523474898, −0.7616885825412951, 0,
0.7616885825412951, 1.541376523474898, 2.108840581088215, 2.770573339394819, 3.284524852629678, 3.770509601440700, 4.415765143102789, 5.073895001712893, 5.383400244029357, 6.125646129454446, 6.454935258051028, 6.942967720268963, 7.359064025777045, 8.082336206689725, 8.475605348638811, 9.205540734438780, 9.343015661540156, 9.790915856005430, 10.46370974138635, 10.82450843045894, 11.24092372526028, 11.84405902601713, 12.23759182303610, 12.89773917210681, 13.17229928775125
