| L(s) = 1 | + 5-s + 2·7-s − 5·13-s + 17-s + 19-s − 6·23-s + 25-s − 9·29-s − 31-s + 2·35-s + 4·37-s + 6·41-s − 2·43-s + 9·47-s − 3·49-s − 9·53-s + 3·59-s + 7·61-s − 5·65-s − 14·67-s − 3·71-s + 11·73-s + 8·79-s + 85-s + 9·89-s − 10·91-s + 95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s − 1.38·13-s + 0.242·17-s + 0.229·19-s − 1.25·23-s + 1/5·25-s − 1.67·29-s − 0.179·31-s + 0.338·35-s + 0.657·37-s + 0.937·41-s − 0.304·43-s + 1.31·47-s − 3/7·49-s − 1.23·53-s + 0.390·59-s + 0.896·61-s − 0.620·65-s − 1.71·67-s − 0.356·71-s + 1.28·73-s + 0.900·79-s + 0.108·85-s + 0.953·89-s − 1.04·91-s + 0.102·95-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
| \(L(1)\) |
\(\approx\) |
\(2.023803735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.023803735\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−14.43416324659471, −14.24944800338576, −13.55355598192290, −13.00187321755018, −12.46516874214368, −12.00923714272781, −11.45586964021490, −10.98276727976941, −10.34190699539698, −9.851205075133409, −9.353924878473955, −8.949970927477059, −7.995810607285962, −7.737767771889117, −7.285732598247310, −6.491718835814105, −5.859475706874503, −5.373078156502567, −4.823002776220885, −4.208843131158588, −3.559016211659384, −2.645433707267892, −2.117071892725948, −1.539769291667489, −0.4861163803870779,
0.4861163803870779, 1.539769291667489, 2.117071892725948, 2.645433707267892, 3.559016211659384, 4.208843131158588, 4.823002776220885, 5.373078156502567, 5.859475706874503, 6.491718835814105, 7.285732598247310, 7.737767771889117, 7.995810607285962, 8.949970927477059, 9.353924878473955, 9.851205075133409, 10.34190699539698, 10.98276727976941, 11.45586964021490, 12.00923714272781, 12.46516874214368, 13.00187321755018, 13.55355598192290, 14.24944800338576, 14.43416324659471
