| L(s) = 1 | − 5-s − 4·7-s − 2·13-s + 17-s − 4·19-s − 8·23-s + 25-s − 6·29-s + 4·35-s + 6·37-s − 2·41-s − 4·43-s − 8·47-s + 9·49-s + 10·53-s + 8·59-s − 6·61-s + 2·65-s − 4·67-s + 8·71-s + 10·73-s + 12·83-s − 85-s + 14·89-s + 8·91-s + 4·95-s + 10·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s − 0.554·13-s + 0.242·17-s − 0.917·19-s − 1.66·23-s + 1/5·25-s − 1.11·29-s + 0.676·35-s + 0.986·37-s − 0.312·41-s − 0.609·43-s − 1.16·47-s + 9/7·49-s + 1.37·53-s + 1.04·59-s − 0.768·61-s + 0.248·65-s − 0.488·67-s + 0.949·71-s + 1.17·73-s + 1.31·83-s − 0.108·85-s + 1.48·89-s + 0.838·91-s + 0.410·95-s + 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.82545353184992, −14.38256186765732, −13.69545252196417, −13.03136802159377, −12.93336928609861, −12.23303267861245, −11.79956382853884, −11.32422574374733, −10.41526326678136, −10.21274889500300, −9.603916948156069, −9.182649099725951, −8.507365655226852, −7.874972170209469, −7.460805062376635, −6.707636516316404, −6.310895280843457, −5.827324972256696, −5.045007721713328, −4.358669059447336, −3.610895625841272, −3.445636278796303, −2.410734370758022, −1.988966971576887, −0.6650192617785987, 0,
0.6650192617785987, 1.988966971576887, 2.410734370758022, 3.445636278796303, 3.610895625841272, 4.358669059447336, 5.045007721713328, 5.827324972256696, 6.310895280843457, 6.707636516316404, 7.460805062376635, 7.874972170209469, 8.507365655226852, 9.182649099725951, 9.603916948156069, 10.21274889500300, 10.41526326678136, 11.32422574374733, 11.79956382853884, 12.23303267861245, 12.93336928609861, 13.03136802159377, 13.69545252196417, 14.38256186765732, 14.82545353184992
